QUARTIC EXERCISES

MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL

Abstract. A correspondence between quartic étale algebras over a field and quadratic étale extensions of cubic étale algebras is set up and investigated. The basic constructions are laid out in general for sets with a profinite group action and for torsors, and translated in terms of étale algebras and Galois algebras. In the final section, a parametrization of cyclic quartic algebras is given.

It is known since the XVIth century that the solution of quartic equations can be obtained by means of auxiliary equations of degree 3, called cubic resolvents. The situation is easily understood in terms of Galois theory. For any integer n ≥ 1, let 𝔖_{n} denote the symmetric group on {1, …, n}. The symmetric group 𝔖_{4} contains a normal subgroup of order 4, Klein’s Vierergruppe

𝔙 = {I, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)},

which is the kernel of the action of 𝔖_{4} on its three Sylow 2-subgroups. Numbering from 1 to 3 these Sylow subgroups, we get an exact sequence of groups

(1)  1 → 𝔙 → 𝔖_{4} ⟶^{ρ} 𝔖_{3} → 1.

Let F be an arbitrary field and P ∈ F [X] be a separable polynomial of degree 4. Let also F_{s} be a separable closure of F and Q ⊂ F_{s} be the subfield generated by the roots of P. The Galois group Gal(Q/F) can be viewed as a subgroup of 𝔖_{4} through its action on the roots of P. The subfield L of Q fixed under Gal(Q/F)∩𝔙 is generated by the roots of a cubic resolvent, as was shown by Lagrange. For a given quartic polynomial P, there are actually many polynomials of degree 3 which qualify as cubic resolvents; only the extension L/F is an invariant of P (or of Q).

Galois cohomology provides another viewpoint on this construction. Since 𝔖_{n} is the automorphism group of the étale F -algebra F^{n} = F × ⋯ × F, it is well-known that the Galois cohomology set H^{1}(F, 𝔖_{n}) is in canonical one-to-one correspondence with the isomorphism classes of étale F -algebras of degree n, see [3, (29.9)]. The map ρ in (1) induces a map

ρ^{1} : H^{1}(F, 𝔖_{4}) → H^{1}(F, 𝔖_{3})

which associates to every quartic étale F -algebra Q a cubic étale F -algebra R(Q) uniquely determined up to isomorphism. If P ∈ F [X] is a separable polynomial of degree 4 with cubic resolvent R, and if Q is the factor algebra Q = F [X]/P, then R(Q) ≃ F [X]/R, see §4.3.

Our first aim is to make explicit the construction of R(Q) from Q. But this construction can be further extended. Each of the three Sylow 2-subgroups of 𝔖_{4} contains two transpositions, and each transposition is in one and only one Sylow subgroup, hence the set of transpositions can be viewed as a double covering of the set of Sylow 2-subgroups. Therefore, the conjugation action of 𝔖_{4} on its six transpositions defines a map

The authors gratefully acknowledge support from the RT Network “K-theory, Linear Algebraic Groups and Related Structures” (contract HPRN-CT-2002-00287). The first author is partially supported by the Swiss National Foundation under grant 21-064712.01. The second author is partially supported by the National Fund for Scientific Research (Belgium).

λ: 𝔖_{4} → 𝔖_{2} ≀ 𝔖_{3},

where the wreath product 𝔖_{2} ≀ 𝔖_{3} is viewed as the group of automorphisms of a double covering of a set of three elements (see §3.1). The map λ extends to an isomorphism of groups

λ̂: 𝔖_{2} × 𝔖_{4} →^{∼} 𝔖_{2} ≀ 𝔖_{3},

see §4.2. The set H^{1}(F, 𝔖_{2} ≀ 𝔖_{3}) classifies the quadratic étale extensions of cubic étale F -algebras (see §3.2), and the induced bijection

λ̂^{1} : H^{1}(F, 𝔖_{2}) × H^{1}(F, 𝔖_{4}) →^{∼} H^{1}(F, 𝔖_{2} ≀ 𝔖_{3})

associates to every pair consisting of a quartic étale F -algebra Q and a quadratic étale F -algebra a quadratic étale extension of the cubic resolvent R(Q). In §4.3, we give an explicit construction of this quadratic extension, and we describe in §4.4 the inverse of λ̂^{1}, attaching a quartic étale F -algebra and a quadratic étale F -algebra to any quadratic étale extension of a cubic étale F -algebra.

In the final sections, we classify quartic étale algebras and their associated quadratic extensions of cubic algebras according to their decomposition into direct products of fields (see §5.3) and we parametrize cyclic quartic extensions.

Contents

1. Γ-sets and coverings 3 1.1. Basic constructions on Γ-sets 3 1.2. Coverings 4 2. Étale algebras and extensions 7 2.1. Basic constructions on étale algebras 8 2.2. Extensions of étale algebras 11 3. Cohomology of permutation groups 14 3.1. Permutations 14 3.2. Cohomology and Γ-sets 14 3.3. Torsors 16 3.4. Cohomology and étale algebras 17 3.5. Galois algebras 18 4. The symmetric group on four elements 20 4.1. Sets of four elements and double coverings 21 4.2. Cohomology 24 4.3. Quartic étale algebras 29 4.4. Quadratic extensions of cubic étale algebras 31 5. Special actions on four elements 36 5.1. Action through 𝔖_{3} 38 5.2. Action through D_{4} 38 5.3. Classification of quartic algebras 43 6. Cyclic quartic algebras 43 6.1. Characteristic not 2 44 6.2. Characteristic 2 47 References 49

La recherche des extensions d’un corps k dont le groupe de Galois sur k est 𝔖_{4} ou 𝔄_{4} n’est pas autre chose, du point de vue des algébristes du XIX^{e} siècle, que la théorie de l’équation du 4^{e} degré. C’est un problème pour lequel ces algébristes n’avaient que du mépris. (A. Weil)

1. Γ-sets and coverings

1.1. Basic constructions on Γ-sets. Let Γ be a profinite group, which will be fixed throughout this section. Finite sets with a continuous action of Γ are called Γ-sets. We let |X| denote the number of elements in a Γ-set X. If X is a Γ-set with n elements, and k is a positive integer, k ≤ n, we let Σ_{k}(X) denote the set of k-tuples of pairwise distinct elements of X and Λ_{k}(X) the set of k-element subsets of X; thus

Σ_{k}(X) = {(ξ_{1}, …, ξ_{k}) ∈ X^{k} | ξ_{i} ≠ ξ_{j} for i ≠ j},
Λ_{k}(X) = {{ξ_{1}, …, ξ_{k}} ⊂ X | ξ_{i} ≠ ξ_{j} for i ≠ j}.

The action of Γ on X induces actions on Σ_{k}(X) and Λ_{k}(X), hence Σ_{k}(X) and Λ_{k}(X) are Γ-sets, and we have

|Σ_{k}(X)| = k!\binom{n}{k}, |Λ_{k}(X)| = \binom{n}{k}.

The symmetric group 𝔖_{k} acts on Σ_{k}(X) by permutation of the entries, and we may consider Λ_{k}(X) as the set of orbits of Σ_{k}(X) under this action, i.e. as the quotient Γ-set

Λ_{k}(X) = Σ_{k}(X)/𝔖_{k}.

For k = n, we may also consider the action of the alternating group 𝔄_{n} on Σ_{n}(X). The quotient is called the discriminant of X and denoted by Δ(X),

Δ(X) = Σ_{n}(X)/𝔄_{n},

see [3, p. 291]. This is a Γ-set with |Δ(X)| = 2 if n ≥ 2.

If n is even, n = 2m, let

γ_{X}: Λ_{m}(X) → Λ_{m}(X)

be the map which associates to every m-element subset of X its complementary subset. Since γ_{X}^{2} = Id, this map defines an action of 𝔖_{2} on Λ_{m}(X). The map γ_{X} is Γ-equivariant (i.e. compatible with the action of Γ), hence the quotient

R(X) = Λ_{m}(X)/𝔖_{2}

is a Γ-set. It is the set of partitions of X into m-element subsets.

Example 1.1. If X = {1, 2, 3, 4}, then

Λ_{2}(X) = {{1, 2}, {3, 4}, {1, 3}, {2, 4}, {1, 4}, {2, 3}}

and

R(X) = {{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}}.

If |X| = 2, the map

γ_{X}: X = Λ_{1}(X) → Λ_{1}(X) = X

interchanges the two elements of X. For X, X′ two Γ-sets with |X| = |X′| = 2, the map

γ_{X} × γ_{X′}: X × X′ → X × X′

defines an action of 𝔖_{2} compatible with the Γ-action. Let

X ∗ X′ = (X × X′)/𝔖_{2},

a Γ-set with |X ∗ X′| = 2. Thus, if X = {x_{1}, x_{2}} and X′ = {x′_{1}, x′_{2}}, then

X ∗ X′ = {{(x_{1}, x′_{1}), (x_{2}, x′_{2})}, {(x_{1}, x′_{2}), (x_{2}, x′_{1})}}.

The following observations are clear:

Proposition 1.2. Let X, X′, be Γ-sets of two elements.

(a) The Γ-action on X ∗ X is trivial.

(b) If the Γ-action on X′ is trivial, then X ∗ X′ ≃ X. (Note that the isomorphism is not canonical.)

(c) The operation ∗ defines a group structure on the set of isomorphism classes of Γ-sets of two elements.

See §3.2 for a cohomological interpretation of the group structure induced by ∗.

1.2. Coverings. A morphism Y ⟵^{π} Z of Γ-sets (i.e. a Γ-equivariant map) is called a covering if the number of elements in each fiber π^{−1}(η) ⊂ Z does not depend on η ∈ Y. This number is called the degree of the covering. Coverings of degree 2 are called double coverings. A morphism of coverings

(Y_{1} ⟵^{π_{1}} Z_{1}) → (Y_{2} ⟵^{π_{2}} Z_{2})

is a pair (σ: Y_{1} → Y_{2}, τ: Z_{1} → Z_{2}) of morphisms such that σ ∘ π_{1} = π_{2} ∘ τ. Given two coverings Y ⟵^{π_{1}} Z_{1} and Y ⟵^{π_{2}} Z_{2} of the same Γ-set Y, an isomorphism over Y is an isomorphism τ: Z_{1} → Z_{2} such that π_{1} = π_{2} ∘ τ.

For any covering Y ⟵^{π} Z let Ω(Z/Y) be the set of sections of π,

Ω(Z/Y) = {{ζ_{1}, …, ζ_{n}} ⊂ Z | {π(ζ_{1}), …, π(ζ_{n})} = Y } ⊂ Λ_{n}(Z).

This is a Γ-set with |Ω(Z/Y)| = d^{n}. If d = 2, the morphism

γ_{Z}: Λ_{n}(Z) → Λ_{n}(Z)

preserves Ω(Z/Y) and induces a 𝔖_{2}-action compatible with the action of Γ. Define

S(Z/Y) = Ω(Z/Y)/𝔖_{2}.

Note that every double covering Y ⟵^{π} Z has an involutive automorphism γ_{Z/Y} which is the identity on Y and interchanges the two elements in each fiber of π. Thus, for {ζ_{1}, …, ζ_{n}} ∈ Ω(Z/Y),

γ_{Z}({ζ_{1}, …, ζ_{m}}) = {γ_{Z/Y} (ζ_{1}), …, γ_{Z/Y} (ζ_{n})}.

Example 1.3. Let Z = {z_{1}, z′_{1}, z_{2}, z′_{2}, z_{3}, z′_{3}}, Y = {1, 2, 3}, and let Y ⟵^{π} Z be the

map which carries z_{i} and z′_{i} to i for i = 1, 2, 3. Then

Ω(Z/Y) = {{z_{1}, z_{2}, z_{3}}, {z_{1}, z′_{2}, z′_{3}}, {z′_{1}, z_{2}, z′_{3}}, {z′_{1}, z′_{2}, z_{3}},
{z′_{1}, z′_{2}, z′_{3}}, {z′_{1}, z_{2}, z_{3}}, {z_{1}, z′_{2}, z_{3}}, {z_{1}, z_{2}, z′_{3}}}

and

S(Z/Y) = {{{z_{1}, z_{2}, z_{3}}, {z′_{1}, z′_{2}, z′_{3}}}, {{z_{1}, z′_{2}, z′_{3}}, {z′_{1}, z_{2}, z_{3}}},
{{z′_{1}, z_{2}, z′_{3}}, {z_{1}, z′_{2}, z_{3}}}, {{z′_{1}, z′_{2}, z_{3}}, {z_{1}, z_{2}, z′_{3}}}}.

Let Y ⟵^{π} Z be an arbitrary double covering of a Γ-set Y of n elements, so |Z| = 2n, and let {ζ_{1}, …, ζ_{n}} ∈ Ω(Z/Y). Even though the n-tuple (ζ_{1}, …, ζ_{n}) is not uniquely determined by the set {ζ_{1}, …, ζ_{n}}, it turns out that the orbit

(ζ_{1}, …, ζ_{n}, γ_{Z/Y} (ζ_{1}), …, γ_{Z/Y} (ζ_{n}))^{𝔄_{2n}}

is well-defined, since every permutation of ζ_{1}, …, ζ_{n} induces a corresponding permutation of γ_{Z/Y} (ζ_{1}), …, γ_{Z/Y} (ζ_{n}), and the resulting permutation of the 2n elements ζ_{1}, …, γ_{Z/Y} (ζ_{n}) is necessarily even. Therefore, we may define

δ({ζ_{1}, …, ζ_{n}}) = (ζ_{1}, …, ζ_{n}, γ_{Z/Y} (ζ_{1}), …, γ_{Z/Y} (ζ_{n}))^{𝔄_{2n}} ∈ Σ_{2n}(Z)/𝔄_{2n} = Δ(Z)

and thus obtain a morphism of Γ-sets

(2)  δ = δ_{Z}: Ω(Z/Y) → Δ(Z).

On the other hand, since S(Z/Y) is a quotient of Ω(Z/Y), there is a canonical map

ε: Ω(Z/Y) → S(Z/Y).

Proposition 1.4. For Y ⟵^{π} Z a double covering with |Y | = n odd, the map

(δ, ε): Ω(Z/Y) → Δ(Z) × S(Z/Y)

is an isomorphism of Γ-sets.

Proof. Since the map (δ, ε) is clearly Γ-equivariant, and since the sets Ω(Z/Y) and Δ(Z) × S(Z/Y) both have 2^{n} elements, it suffices to show that (δ, ε) is injective. Suppose {ζ_{1}, …, ζ_{n}}, {ζ′_{1}, …, ζ′_{n}} ∈ Ω(Z/Y) are distinct elements such that ε({ζ_{1}, …, ζ_{n}}) = ε({ζ′_{1}, …, ζ′_{n}}); then {ζ′_{1}, …, ζ′_{n}} = γ_{Z}({ζ_{1}, …, ζ_{n}}) and we may assume the elements are numbered in such a way that π(ζ′_{i}) = π(ζ_{i}) (i.e. ζ′_{i} = γ_{Z/Y} (ζ_{i})) for i = 1, …, n. Since the permutation which interchanges ζ_{i} and ζ′_{i} for i = 1, …, n is odd, we have

(ζ_{1}, …, ζ_{n}, ζ′_{1}, …, ζ′_{n})^{𝔄_{2n}} ≠ (ζ′_{1}, …, ζ′_{n}, ζ_{1}, …, ζ_{n})^{𝔄_{2n}},

hence δ({ζ_{1}, …, ζ_{n}}) ≠ δ({ζ′_{1}, …, ζ′_{n}}). □

If Y ⟵^{π} Z and Y ⟵^{π′} Z′ are two double coverings of the same Γ-set Y with n elements, consider the fiber product

Z ×_{Y} Z′ = {(ζ, ζ′) ∈ Z × Z′ | π(ζ) = π′(ζ′)} ⊂ Z × Z′.

The group 𝔖_{2} acts on Z ×_{Y} Z′ by mapping (ζ, ζ′) to (γ_{Z/Y} (ζ), γ_{Z′/Y} (ζ′)). Let

Z ∗_{Y} Z′ = (Z ×_{Y} Z′)/𝔖_{2}.

The canonical map Y ⟵^{π×π′} Z ×_{Y} Z′ induces a map

Y ⟵^{π∗π′} Z ∗_{Y} Z′

which is a double covering. In particular, for any Γ-set X of two elements and any covering Y ⟵^{π} Z of degree 2, we may consider

Y ⟵^{π_{2}∗π} (X × Y) ∗_{Y} Z

where Y ⟵^{π_{2}} X × Y is the projection map. Abusing notation, we write simply

Y ⟵^{π} X ∗ Z

for this double covering. The proof of the following easy proposition is omitted:

Proposition 1.5. Let X be a set of two elements with trivial Γ-action, and let Y ⟵^{π} Z be a double covering.

(a) The covering Y ⟵^{π∗π} Z ∗_{Y} Z is isomorphic to Y ⟵^{π_{2}} X × Y.

(b) The covering Y ⟵^{π} X ∗ Z is (non-canonically) isomorphic to Y ⟵^{π} Z.

(c) The operation ∗_{Y} defines a group structure on the set of isomorphism classes over Y of double coverings of Y. The neutral element is the isomorphism class of Y ⟵^{π_{2}} X × Y.

Proposition 1.6. Let Y ⟵^{π} Z and Y ⟵^{π} Z′ be two double coverings. There is a canonical isomorphism Δ(Z ∗_{Y} Z′) ≃ Δ(Z) ∗ Δ(Z′).

Proof. Recall from (2) the map

δ_{Z}: Ω(Z/Y) → Δ(Z)

defined for any double covering Y ⟵^{π} Z of a Γ-set Y with |Y | = n. In the sequel, we write simply δ_{Z}(z_{1}, …, z_{n}) for δ_{Z}({z_{1}, …, z_{n}}). For ω, ω′ ∈ Ω(Z/Y), we have δ_{Z}(ω) = δ_{Z}(ω′) if and only if |ω ∩ ω′| ≡ n mod 2. In particular, the map δ_{Z} is onto.

Let Y ⟵^{π} Z and Y ⟵^{π′} Z′ be two double coverings of Y. Denote simply by the canonical automorphisms γ_{Z/Y} and γ_{Z′/Y} and also the canonical automorphisms of Δ(Z) and Δ(Z′). For {z_{1}, …, z_{n}} ∈ Ω(Z/Y) and {z′_{1}, …, z′_{n}} ∈ Ω(Z′/Y), we have

δ_{Z}(\overline{z_{1}}, z_{2}, …, z_{n}) = \overline{δ_{Z}(z_{1}, z_{2}, …, z_{n})}

and, similarly,

δ_{Z′}(\overline{z′_{1}}, z′_{2}, …, z′_{n}) = \overline{δ_{Z′}(z′_{1}, z′_{2}, …, z′_{n})}.

Therefore, the element

\{(δ_{Z}(z_{1}, …, z_{n}), δ_{Z′}(z′_{1}, …, z′_{n})), (\overline{δ_{Z}(z_{1}, …, z_{n})}, \overline{δ_{Z′}(z′_{1}, …, z′_{n})})\} ∈ Δ(Z)∗Δ(Z′)

depends only on

ω = \{\{(z_{1}, z′_{1}), (\overline{z_{1}}, \overline{z′_{1}})\}, …, \{(z_{n}, z′_{n}), (\overline{z_{n}}, \overline{z′_{n}})\}\} ∈ Ω(Z ∗_{Y} Z′/Y).

We thus have a canonical map

ψ: Ω(Z ∗_{Y} Z′/Y) → Δ(Z) ∗ Δ(Z′).

If ω′ ∈ Ω(Z ∗_{Y} Z′/Y) is obtained from ω by substituting

\{(z_{1}, \overline{z′_{1}}), (\overline{z_{1}}, z′_{1})\} for \{(z_{1}, z′_{1}), (\overline{z_{1}}, \overline{z′_{1}})\},

then ψ(ω) ≠ ψ(ω′), hence ψ is onto. On the other hand, if ω′ is obtained from ω by an even number of changes as above, then ψ(ω) = ψ(ω′). Therefore, ψ(ω) = ψ(ω′) if |ω ∩ ω′| ≡ n mod 2, and it follows that ψ factors through the map

δ_{Z∗_{Y} Z′}: Ω(Z ∗_{Y} Z′/Y) → Δ(Z ∗_{Y} Z′).

This completes the proof. □

For later use, we record another case where the map δ of (2) can be used in the computation of a discriminant. Let X be a Γ-set of two elements. For any Γ-set Y, we may consider the double covering Y ⟵^{π_{2}} X × Y given by the projection.

Proposition 1.7. If |X| = 2 and |Y | is odd, the composition of the Γ-equivariant map

ι: X → Ω(X × Y/Y), ι(x) = {(x, y) | y ∈ Y }

and

δ_{X×Y} : Ω(X × Y/Y) → Δ(X × Y)

defines an isomorphism

δ ∘ ι: X →^{∼} Δ(X × Y).

Proof. Since |Y | is odd, we have

δ ∘ ι(γ_{X}(x)) = γ_{Δ(X×Y)}(δ ∘ ι(x)) for x ∈ X.

Therefore, δ ∘ ι is surjective, hence bijective. □

Proposition 1.8. Let X be a Γ-set of two elements, and let Y ⟵^{π} Z be a double covering. There is a canonical isomorphism

S(Z/Y) ≃ S(X ∗ Z/Y).

Proof. For simplicity of notation, let X = {+, −} and denote by the canonical automorphism γ_{Z/Y}. We may then identify X∗Z with the set of formal polynomials ζ − \overline{ζ}, for ζ ∈ Z. Note however that the Γ-action on these polynomials is not linear, since Γ may act non trivially on {+, −}. The structure map Y ← X ∗ Z carries ζ − \overline{ζ} to π(ζ) = π(\overline{ζ}). Therefore,

Ω(X ∗ Z/Y) = \{\{ζ_{1} − \overline{ζ_{1}}, …, ζ_{n} − \overline{ζ_{n}}\} | \{π(ζ_{1}), …, π(ζ_{n})\} = Y \}

and

S(X ∗ Z/Y) =

\{\{\{ζ_{1} − \overline{ζ_{1}}, …, ζ_{n} − \overline{ζ_{n}}\}, \{\overline{ζ_{1}} − ζ_{1}, …, \overline{ζ_{n}} − ζ_{n}\}\} | \{π(ζ_{1}), …, π(ζ_{n})\} = Y \}.

On the other hand,

S(Z/Y) = \{\{\{ζ_{1}, …, ζ_{n}\}, \{\overline{ζ_{1}}, …, \overline{ζ_{n}}\}\} | \{π(ζ_{1}), …, π(ζ_{n})\} = Y \}.

The map

\{\{ζ_{1}, …, ζ_{n}\}, \{\overline{ζ_{1}}, …, \overline{ζ_{n}}\}\} ↦ \{\{ζ_{1} − \overline{ζ_{1}}, …, ζ_{n} − \overline{ζ_{n}}\}, \{\overline{ζ_{1}} − ζ_{1}, …, \overline{ζ_{n}} − ζ_{n}\}\}

is a Γ-isomorphism S(Z/Y) →^{∼} S(X ∗ Z/Y), whatever the action of Γ on {+, −} is.

□

2. Étale algebras and extensions

In this section, F is an arbitrary field. We denote by F_{s} a separable closure of F and let Γ = Gal(F_{s}/F). A finite-dimensional F -algebra E is called étale if E ⊗_{F} F_{s} is isomorphic to a split F_{s}-algebra F_{s} × ⋯ × F_{s}. For any étale F -algebra E of dimension n, the set of F -algebra homomorphisms

X(E) = Hom_{F -alg}(E, F_{s})

is a Γ-set of n elements since Γ acts on F_{s}.

Conversely, starting from a Γ-set X with |X| = n, we may let Γ act by semi-linear automorphisms on the F_{s}-algebra of maps Map(X, F_{s}). The fixed F -algebra

M(X) = Map(X, F_{s})^{Γ} = {f: X → F_{s} | γf(ξ) = f(^{γ}ξ) for γ ∈ Γ, ξ ∈ X}

is étale of dimension n. Moreover, there are canonical isomorphisms

M(X(E)) ≃ E, X(M(X)) ≃ X

(see [3, (18.19)]), so that the functors M and X define an anti-equivalence between the category Ét_{F} of étale F -algebras (with F -algebra homomorphisms) and the category Set_{Γ} of Γ-sets. Under this anti-equivalence, the direct product (resp. tensor product) of F -algebras corresponds to the disjoint union (resp. direct product) of Γ-sets: for E_{1}, E_{2} étale F -algebras, there are obvious identifications

X(E_{1} ⊗ E_{2}) = X(E_{1}) × X(E_{2}) and X(E_{1} × E_{2}) = X(E_{1}) ∐ X(E_{2}).

Moreover, if G is a group acting on an étale F -algebra E by F -automorphisms, then for the fixed subalgebra E^{G} we have

X(E^{G}) = X(E)/G

since E^{G} is the equalizer of the automorphisms σ: E → E for σ ∈ G, and X(E)/G is the co-equalizer of the corresponding automorphisms of X(E). Through the antiequivalence Ét_{F} ≡ Set_{Γ}, the constructions on Γ-sets defined in §1.1 therefore have counterparts in the category of étale F -algebras. The aim of this section is to make them explicit.

2.1. Basic constructions on étale algebras. Let E be an étale F -algebra of dimension n. Under the canonical isomorphism E ≃ M(X(E)), the idempotents of E correspond to the characteristic functions of Γ-subsets of X(E). If e ∈ E is the characteristic function of a subset Y ⊂ X(E), then multiplication by e defines an isomorphism E/(1 − e)E →^{∼} eE. Moreover, X(eE) = Y and under the anti-equivalence Ét_{F} ≡ Set_{Γ}, the map E → eE corresponds to the inclusion X(E) ← Y.

Example 2.1. If E is the split étale F -algebra E = F^{n}, then X(E) is in duality with the set e_{1}, …, e_{n} of minimal idempotents of E, namely X(E) = {ξ_{1}, …, ξ_{n}} where

ξ_{i}(e_{j}) = \begin{cases}0 & if i ≠ j, \\ 1 & if i = j. \end{cases}

The idempotent corresponding to a subset Y = {ξ_{i} | i ∈ I} ⊂ X(E) is ∑_{i∈I} e_{i}.

Let E be an arbitrary étale F -algebra of dimension n. For any integer k with 1 ≤ k ≤ n, we let s_{k} ∈ E^{⊗k} be the idempotent corresponding to the characteristic function of the subset

Σ_{k}(X(E)) = {(ξ_{1}, …, ξ_{k}) | ξ_{i} ≠ ξ_{j} for i ≠ j} ⊂ X(E)^{k} = X(E^{⊗k}).

Therefore, letting Σ_{k}(E) = s_{k}E^{⊗k}, we have

X(Σ_{k}(E)) = Σ_{k}(X(E)).

In particular, for k = 2 the idempotent 1 − s_{2} is the characteristic function of the diagonal of X(E) × X(E) = X(E ⊗ E). It is the separability idempotent of E, see [3, p. 285]. For k ≥ 3, the idempotent s_{k} can also be defined in terms of the separability idempotent of E, see [8, p. 42], [3, p. 320].

The symmetric group 𝔖_{k} acts on E^{⊗k} by permutation of the factors, and the idempotent s_{k} is fixed under this action, so 𝔖_{k} also acts on Σ_{k}(E). We consider the fixed subalgebra

Λ_{k}(E) = Σ_{k}(E)^{𝔖_{k}} = s_{k}(E^{⊗k})^{𝔖_{k}}.

We have

X(Λ_{k}(E)) = Λ_{k}(X(E))

since under the anti-equivalence Ét_{F} ≡ Set_{Γ} the fixed algebra under 𝔖_{k} corresponds to the factor set under the 𝔖_{k}-action.

The discriminant of E is defined by

Δ(E) = Σ_{n}(E)^{𝔄_{n}},

and we have

X(Δ(E)) = Δ(X(E)).

Example 2.2. If E = F^{n} is split with minimal idempotents e_{1}, …, e_{n}, then

s_{k} = ∑_{(i_{1},...,i_{k})} e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{k}}

where (i_{1}, …, i_{k}) ranges over the k-tuples of pairwise distinct integers from 1 to n (i.e. (i_{1}, …, i_{k}) ∈ Σ_{k}({1, …, n})). The algebra Σ_{k}(E) is then split, with minimal idempotents

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{k}}

for (i_{1}, …, i_{k}) ∈ Σ_{k}({1, …, n}). Similarly, the algebra Λ_{k}(E) is split, with minimal idempotents

e_{{i_{1},...,i_{k}}} = ∑_{σ∈𝔖_{k}} e_{i_{σ(1)}} ⊗ ⋯ ⊗ e_{i_{σ(k)}}

for {i_{1}, …, i_{k}} a subset of k elements of {1, …, n}, i.e. {i_{1}, …, i_{k}} ∈ Λ_{k}({1, …, n}). We also have Δ(E) ≃ F × F if n ≥ 2, with minimal idempotents

∑_{σ∈𝔄_{n}} e_{σ(1)} ⊗ ⋯ ⊗ e_{σ(n)}  and  ∑_{σ∉𝔄_{n}} e_{σ(1)} ⊗ ⋯ ⊗ e_{σ(n)}.

For an arbitrary étale F -algebra E of dimension n, the algebra Λ_{k}(E) can also be viewed as an algebra of linear transformations of the exterior power ⋀^{k} E (where E is just regarded as a vector space), as we now show.

Multiplication on the left defines an F -algebra homomorphism (the regular representation)

E^{⊗k} → End_{F} (E^{⊗k}).

As pointed out by Saltman [7, Lemma 1.1], the image of (E^{⊗k})^{𝔖_{k}} in End_{F} (E^{⊗k}) preserves the kernel of the canonical map E^{⊗k} → ⋀^{k} E. Therefore, there is an induced F -algebra homomorphism

(3)  (E^{⊗k})^{𝔖_{k}} → End_{F} (⋀^{k} E).

Lemma 2.3. The homomorphism (3) maps s_{k} to the identity map on ⋀^{k} E.

Proof. It suffices to check the assertion over an extension of F. We may thus assume that E is split, E = F^{n}. Let e_{1}, …, e_{n} be the minimal idempotents of E; then s_{k} is as in Example 2.2 and its image in End_{F} (⋀^{k} E) maps e_{j_{1}} ∧ … ∧ e_{j_{k}} to

∑_{(i_{1},...,i_{k})} e_{i_{1}}e_{j_{1}} ∧ … ∧ e_{i_{k}}e_{j_{k}} = e_{j_{1}} ∧ … ∧ e_{j_{k}}.

□

In view of the lemma, the homomorphism (3) induces an F -algebra homomorphism

φ_{k}: (E^{⊗k})^{𝔖_{k}}/(1 − s_{k})(E^{⊗k})^{𝔖_{k}} = Λ_{k}(E) → End_{F} (⋀^{k} E).

Saltman [7, Lemma 1.3] has shown that the image of this map has dimension \binom{n}{k} = dim Λ_{k}(E), hence φ_{k} is injective.

For instance, for a, x_{1}, …, x_{k} ∈ E,

φ_{k}(s_{k}(a ⊗ ⋯ ⊗ a))(x_{1} ∧ … ∧ x_{k}) = ax_{1} ∧ … ∧ ax_{k},

and

φ_{k}(s_{k}(a ⊗ 1 ⊗ ⋯ ⊗ 1 + 1 ⊗ a ⊗ ⋯ ⊗ 1 + ⋯ + 1 ⊗ ⋯ ⊗ 1 ⊗ a))(x_{1} ∧ … ∧ x_{k}) =

(ax_{1} ∧ x_{2} ∧ … ∧ x_{k}) + (x_{1} ∧ ax_{2} ∧ … ∧ x_{k}) + ⋯ + (x_{1} ∧ x_{2} ∧ … ∧ ax_{k}).

Now, consider the case where n is even, n = 2m. Since dim ⋀^{n} E = 1, and the exterior product ⋀^{m} E × ⋀^{m} E → ⋀^{n} E is a nonsingular bilinear pairing, there is an adjoint involution γ on End_{F} (⋀^{m} E), defined by the equation

f(x) ∧ y = x ∧ γ(f)(y) for x, y ∈ ⋀^{m} E, f ∈ End_{F} (⋀^{m} E).

Proposition 2.4. The involution γ preserves the image of φ_{k}. Therefore, there is an induced involutive automorphism γ_{E} on Λ_{m}(E) defined by φ_{m} ∘ γ_{E} = γ ∘ φ_{m}.

Proof. Extending scalars to a separable closure, we may assume E is split. It is then spanned by its minimal idempotents e_{1}, …, e_{n}, and Λ_{m}(E) is spanned by the minimal idempotents e_{{i_{1},...,i_{m}}} defined in Example 2.2 for {i_{1}, …, i_{m}} ∈ Λ_{m}({1, …, n}).

Computation shows that for {i_{1}, …, i_{m}}, {j_{1}, …, j_{m}} ∈ Λ_{m}({1, …, n}),

φ_{m}(e_{\{i_{1},...,i_{m}\}})(e_{j_{1}} ∧ … ∧ e_{j_{m}}) = \begin{cases}e_{j_{1}} ∧ … ∧ e_{j_{m}} & if \{i_{1}, …, i_{m}\} = \{j_{1}, …, j_{m}\}, \\ 0 & if \{i_{1}, …, i_{m}\} ≠ \{j_{1}, …, j_{m}\}. \end{cases}

It is then easily verified that

(4)  γ ∘ φ_{m}(e_{{i_{1},...,i_{m}}}) = e_{{k_{1},...,k_{m}}},

where {k_{1}, …, k_{m}} is the complementary subset of {i_{1}, …, i_{m}} in {1, …, n}. □

We may now define

R(E) = Λ_{m}(E)^{𝔖_{2}} = {x ∈ Λ_{m}(E) | γ_{E}(x) = x}.

Theorem 2.5. X(R(E)) = R(X(E)).

Proof. It suffices to see that under the anti-equivalence Ét_{F} ≡ Set_{Γ}, the automorphism γ_{E} of Λ_{m}(E) corresponds to the permutation γ_{X(E)} of X(Λ_{m}(E)) = Λ_{m}(X(E)). Again, we may extend scalars to a separable closure of F and assume E is split. Using the same notation as in the preceding proof, we may identify X(E) with the dual basis of e_{1}, …, e_{n}. Equation (4) shows that

γ_{E}(e_{{i_{1},...,i_{m}}}) = e_{{k_{1},...,k_{m}}},

where {k_{1}, …, k_{m}} is the complementary subset of {i_{1}, …, i_{m}} in {1, …, n}, and the proof is complete. □

When dim E = 2, the algebra E is called a quadratic étale F -algebra. In the notation above, we then have m = 1, so Λ_{m}(E) = E, hence E carries a canonical involutive automorphism γ_{E}. Let E, E′ be two quadratic étale F -algebras, with canonical involutive automorphisms γ_{E}, γ_{E′}. The tensor product γ_{E} ⊗ γ_{E′} defines a 𝔖_{2}-action on E ⊗_{F} E′, and we let

E ∗ E′ = (E ⊗_{F} E′)^{𝔖_{2}}.

Proposition 2.6. X(E ∗ E′) = X(E) ∗ X(E′).

Proof.

X((E ⊗_{F} E′)^{𝔖_{2}}) = X(E ⊗_{F} E′)/𝔖_{2} = (X(E) × X(E′))/𝔖_{2}.
□

Let Quad(F) be the set of isomorphism classes of quadratic étale F -algebras, which is in bijection under X with the set of isomorphism classes of Γ-sets of two elements. The following analogue of Proposition 1.2 is easily proved, either directly or by reduction to Proposition 1.2 under the anti-equivalence Ét_{F} ≡ Set_{Γ}:

Proposition 2.7. Let E, E′ be two quadratic étale F -algebras.

(a) The F -algebra E ∗ E is split: E ∗ E ≃ F × F.

(b) If the algebra E′ is split, then E ∗ E′ ≃ E (not canonically).

(c) The operation ∗ defines a group structure on the set Quad(F).

2.2. Extensions of étale algebras. An étale F -algebra B containing an F -algebra A (necessarily étale) is called an extension of degree d of A if it is a free A-module of rank d. Equivalently, this condition means that the inclusion A ⟶^{i} B corresponds under the anti-equivalence Ét_{F} ≡ Set_{Γ} to a map X(A) ⟵^{i^{∗}} X(B) which is a covering of degree d. Extensions of degree 2 are called quadratic extensions.

Suppose B/A is an extension of degree d. Let dim_{F} A = n (hence dim_{F} B = nd), and let s_{n}^{A} ∈ (A^{⊗n})^{𝔖_{n}} be the idempotent defining Σ_{n}(A), see §2.1. Then i^{⊗n}(s_{n}^{A}) is an idempotent of (B^{⊗n})^{𝔖_{n}}. Define

Ω(B/A) = i^{⊗n}(s_{n}^{A}) · (B^{⊗n})^{𝔖_{n}}.

Proposition 2.8. There is a canonical surjective map Λ_{n}(B) → Ω(B/A), and

X(Ω(B/A)) = Ω(X(B)/ X(A)).

Therefore, dim_{F} Ω(B/A) = d^{n}.

Proof. The set X(Ω(B/A)) is the set of F -algebra homomorphisms (B^{⊗n})^{𝔖_{n}} → F_{s} which map i^{⊗n}(s_{n}^{A}) to 1. Since

X((B^{⊗n})^{𝔖_{n}}) = X(B)^{n}/𝔖_{n},

every such homomorphism is the orbit of an n-tuple (ξ_{1}, …, ξ_{n}) of elements of X(B); the condition that the homomorphism maps i^{⊗n}(s_{n}^{A}) to 1 is equivalent to the following: the homomorphism (A^{⊗n})^{𝔖_{n}} → F_{s} associated to the n-tuple (i^{∗}(ξ_{1}), …, i^{∗}(ξ_{n})) maps s_{n}^{A} to 1. In view of the definition of s_{n}^{A}, this means that i^{∗}(ξ_{1}), …, i^{∗}(ξ_{n}) are pairwise distinct, hence

{i^{∗}(ξ_{1}), …, i^{∗}(ξ_{n})} = X(A)

since | X(A)| = n. Of course, this condition implies that ξ_{1}, …, ξ_{n} are pairwise distinct, hence {ξ_{1}, …, ξ_{n}} ∈ Λ_{n}(X(B)). Thus,

X(Ω(B/A)) = {{ξ_{1}, …, ξ_{n}} ⊂ X(B) | {i^{∗}(ξ_{1}), …, i^{∗}(ξ_{n})} = X(A)}

= Ω(X(B)/ X(A)).

The inclusion Ω(X(B)/ X(A)) ⊂ Λ_{n}(X(B)) yields the canonical surjective map Λ_{n}(B) → Ω(B/A) under the anti-equivalence Ét_{F} ≡ Set_{Γ}. □

Now, suppose d = 2, so that dim_{F} B = 2n. The canonical involutive automorphism γ_{X(B)/ X(A)} on X(B) corresponds to a canonical involutive automorphism γ_{B/A} of B such that

A = {x ∈ B | γ_{B/A}(x) = x}.

On the other hand, there is also a “complementary subset” map

γ_{X(B)} : Λ_{n}(X(B)) → Λ_{n}(X(B)).

Since this map preserves Ω(X(B)/ X(A)), the corresponding map γ_{B}: Λ_{n}(B) → Λ_{n}(B) induces an involutive automorphism on Ω(B/A), which we also denote by γ_{B}, and we may consider the subalgebra of fixed points

S(B/A) = Ω(B/A)^{𝔖_{2}} = {x ∈ Ω(B/A) | γ_{B}(x) = x}.

By definition, it is clear that

X(S(B/A)) = S(X(B)/ X(A)),

hence dim_{F} S(B/A) = 2^{n−1}.

Example 2.9. Suppose A and B are split of dimensions 3 and 6 respectively, with minimal idempotents e_{1}, e_{2}, e_{3} and f_{1}, f′_{1}, f_{2}, f′_{2}, f_{3}, f′_{3} such that

e_{i} = f_{i} + f′_{i} for i = 1, 2, 3.

As observed in Example 2.2,

s_{3}^{A} = ∑_{σ∈𝔖_{3}} e_{σ(1)} ⊗ e_{σ(2)} ⊗ e_{σ(3)},

and {e_{σ(1)} ⊗ e_{σ(2)} ⊗ e_{σ(3)} | σ ∈ 𝔖_{3}} is the set of minimal idempotents of Σ_{3}(A). Denoting in general by ∑ u ⊗ v ⊗ w the sum of the six products obtained by permuting the factors u, v, w (so

∑ u ⊗ v ⊗ w = ∑_{σ∈𝔖_{3}} u_{σ(1)} ⊗ u_{σ(2)} ⊗ u_{σ(3)}

where u_{1} = u, u_{2} = v, u_{3} = w), the minimal idempotents of Ω(B/A) are

g_{0} = ∑ f_{1} ⊗ f_{2} ⊗ f_{3},  g′_{0} = ∑ f′_{1} ⊗ f′_{2} ⊗ f′_{3},
g_{1} = ∑ f_{1} ⊗ f′_{2} ⊗ f′_{3},  g′_{1} = ∑ f′_{1} ⊗ f_{2} ⊗ f_{3},
g_{2} = ∑ f′_{1} ⊗ f_{2} ⊗ f′_{3},  g′_{2} = ∑ f_{1} ⊗ f′_{2} ⊗ f_{3},
g_{3} = ∑ f′_{1} ⊗ f′_{2} ⊗ f_{3},  g′_{3} = ∑ f_{1} ⊗ f_{2} ⊗ f′_{3}.

The involution γ_{B} interchanges g_{i} and g′_{i} for i = 0, …, 3, hence the minimal idempotents of S(B/A) are

g_{0} + g′_{0},  g_{1} + g′_{1},  g_{2} + g′_{2},  g_{3} + g′_{3}.

Let B, B′ be quadratic extensions of an étale F -algebra A. The canonical map B ⊗_{F} B′ → B ⊗_{A} B′ induces an injective map

X(B) × X(B′) = X(B ⊗_{F} B′) ← X(B ⊗_{A} B′)

which identifies X(B ⊗_{A} B′) to the fiber product X(B) ×_{X(A)} X(B′). The tensor product γ_{B/A} ⊗ γ_{B′/A} defines an action of 𝔖_{2} on B ⊗_{A} B′ by A-automorphisms, and we let

B ∗_{A} B′ = (B ⊗_{A} B′)^{𝔖_{2}}.

The following result is clear:

Proposition 2.10. X(B ∗_{A} B′) = X(B) ∗_{X(A)} X(B′).

If E is an étale F -algebra of dimension 2, then E ⊗_{F} A is a quadratic extension of A. For any quadratic extension B/A, we have E ⊗_{F} B = (E ⊗_{F} A) ⊗_{A} B and we write simply

E ∗ B  for  (E ⊗_{F} A) ∗_{A} B.

Let Quad(A) be the set of isomorphism classes over A of quadratic extensions of A. The following proposition is the analogue of Proposition 1.5:

Proposition 2.11. Let E be a split étale F -algebra of dimension 2 (i.e. E ≃ F ×F), and let B/A be a quadratic extension of étale F -algebras.

(a) The extension (B ∗_{A} B)/A is isomorphic to (E ⊗_{F} A)/A (hence also to (A × A)/A).

(b) The extension (E ∗ B)/A is (non-canonically) isomorphic to B/A.

(c) The operation ∗_{A} defines a group structure on Quad(A). The neutral element is the isomorphism class of (A × A)/A.

It is clear that Propositions 1.4, 1.6 and 1.8 have analogues for étale algebras. We record them below.

Proposition 2.12. Let E be an étale F -algebra of dimension 2, and let B/A and B′/A be quadratic extensions of an étale F -algebra A. There are canonical isomorphisms:

(a) Δ(B) ⊗ S(B/A) ≃ Ω(B/A) (if dim A is odd);

(b) Δ(B ∗_{A} B′) ≃ Δ(B) ∗ Δ(B′);
(c) S(E ∗ B/A) ≃ S(B/A).

3. Cohomology of permutation groups

3.1. Permutations. For any finite set X, let 𝔖_{X} be the symmetric group of X, i.e. the group of all permutations of X. Thus, 𝔖_{X} = 𝔖_{n} for X = {1, …, n}. Every permutation of a set X of n elements induces a permutation of the sets Σ_{k}(X), Λ_{k}(X) (for k ≤ n), Δ(X), and of R(X) if n is even. There are therefore canonical group homomorphisms

𝔖_{X} → 𝔖_{Σ_{k}(X)}, 𝔖_{X} → 𝔖_{Λ_{k}(X)}, 𝔖_{X} → 𝔖_{Δ(X)}, 𝔖_{X} → 𝔖_{R(X)} (if n is even).

(If n ≥ 2, the map 𝔖_{X} ⟶^{sgn} 𝔖_{Δ(X)} = 𝔖_{2} is the signature map.)

If Y ⟵^{π} Z is a covering of degree d of a set of n elements, let

𝔖_{Z/Y} = {(σ, τ) ∈ 𝔖_{Y} × 𝔖_{Z} | π ∘ τ = σ ∘ π}

be the group of automorphisms of the covering. The map (σ, τ) ↦ τ identifies 𝔖_{Z/Y} to a subgroup of 𝔖_{Z}. On the other hand, the map (σ, τ) ↦ σ defines a surjective homomorphism

β_{Z/Y} : 𝔖_{Z/Y} → 𝔖_{Y}

whose kernel is isomorphic to 𝔖_{d}^{n} upon identifying each fiber of π with {1, …, d}. Therefore, the group 𝔖_{Z/Y} has order (d!)^{n}n! and can be identified to a wreath product

𝔖_{Z/Y} ≃ 𝔖_{d} ≀ 𝔖_{n}.

Automorphisms of the covering Y ← Z induce permutations of Ω(Z/Y), and of S(Z/Y) if d = 2, hence there are canonical group homomorphisms

ω_{Z/Y} : 𝔖_{Z/Y} → 𝔖_{Ω(Z/Y)}, s_{Z/Y} : 𝔖_{Z/Y} → 𝔖_{S(Z/Y)} if d = 2.

For later use, note that the kernel of s_{Z/Y} is the “diagonal” subgroup 𝔖_{2} of 𝔖_{Z/Y}, whose nontrivial element is γ_{Z/Y}. This diagonal subgroup is central in 𝔖_{Z/Y}.

On the other hand, every permutation of a set X with n = 2m elements induces an automorphism of the covering R(X) ⟵^{ε} Λ_{m}(X), hence there is a canonical group homomorphism

λ_{X}: 𝔖_{X} → 𝔖_{Λ_{m}(X)/ R(X)} ⊂ 𝔖_{Λ_{m}(X)}.

Proposition 3.1. If m ≥ 2, the image of λ_{X} is in the kernel of the signature map

sgn: 𝔖_{Λ_{m}(X)} → 𝔖_{Δ(Λ_{m}(X))}.

Moreover, the composition of λ_{X} and the canonical homomorphism s_{Λ_{m}(X)/ R(X)} is an injective map

𝔖_{X} ↪ 𝔖_{S(Λ_{m}(X)/ R(X))}.

The proof is left to the reader.

3.2. Cohomology and Γ-sets. As in §1, we denote by Γ a profinite group, which will be fixed throughout this subsection. The action of Γ on a set X with |X| = n can be viewed as a group homomorphism

Γ → 𝔖_{X} ≃ 𝔖_{n}.

Since the isomorphism 𝔖_{X} ≃ 𝔖_{n} depends on the indexing of the elements in X, the homomorphism Γ → 𝔖_{n} is defined by X up to conjugation by an element in 𝔖_{n}. Therefore, there is a canonical one-to-one correspondence between isomorphism classes of Γ-sets of n elements and the cohomology set H^{1}(Γ, 𝔖_{n}) (with the trivial action of Γ on 𝔖_{n}), by definition of this cohomology set. Under this correspondence, the distinguished element of H^{1}(Γ, 𝔖_{n}) is mapped to the set with trivial Γ-action.

Since the symmetric group 𝔖_{2} is abelian, there is an abelian group structure on H^{1}(Γ, 𝔖_{2}). We leave it to the reader to verify that the product of the isomorphism classes of the Γ-sets X, X′ with |X| = |X′| = 2 is the isomorphism class of X ∗ X′.

Similarly, every covering Y ⟵^{π} Z of degree d of a Γ-set Y with |Y | = n yields a group homomorphism

Γ → 𝔖_{Z/Y} ≃ 𝔖_{d} ≀ 𝔖_{n},

and there is a canonical one-to-one correspondence between isomorphism classes of coverings of degree d of Γ-sets of n elements and the cohomology set H^{1}(Γ, 𝔖_{d}≀𝔖_{n}), which maps the distinguished element of the cohomology set to the covering with trivial Γ-action.

The basic constructions in subsections 1.1 and 1.2 yield canonical maps of cohomology sets through the induced homomorphisms of permutation groups (see §3.1). For instance, if X is a Γ-set of n elements and k ≤ n, the canonical homomorphism σ_{k}: 𝔖_{X} → 𝔖_{Σ_{k}(X)} induces a morphism of pointed sets

σ_{k}^{1} : H^{1}(Γ, 𝔖_{X}) → H^{1}(Γ, 𝔖_{Σ_{k}(X)}).

Since the Γ-action on Σ_{k}(X) is induced by the Γ-action on X through σ_{k}, the morphism σ_{k}^{1} maps the isomorphism class of X to the isomorphism class of Σ_{k}(X). A similar statement obviously holds for the morphisms

H^{1}(Γ, 𝔖_{X}) → H^{1}(Γ, 𝔖_{Λ_{k}(X)}),
H^{1}(Γ, 𝔖_{X}) → H^{1}(Γ, 𝔖_{Δ(X)}),
H^{1}(Γ, 𝔖_{X}) → H^{1}(Γ, 𝔖_{R(X)}) if n is even.

Similarly, if Y ⟵^{π} Z is a covering of degree d of Γ-sets, the canonical homomorphisms ω_{Z/Y} and s_{Z/Y} of §3.1 induce morphisms of pointed sets

ω_{Z/Y}^{1} : H^{1}(Γ, 𝔖_{Z/Y}) → H^{1}(Γ, 𝔖_{Ω(Z/Y)}),
s_{Z/Y}^{1} : H^{1}(Γ, 𝔖_{Z/Y}) → H^{1}(Γ, 𝔖_{S(Z/Y)}) if d = 2.

Since the Γ-action on Ω(Z/Y) and S(Z/Y) (if d = 2) are induced by the Γ-action on Z/Y through ω_{Z/Y} and σ_{Z/Y} respectively, the morphisms ω_{Z/Y}^{1} and s_{Z/Y}^{1} map the isomorphism class of the covering Z/Y to the isomorphism class of the Γ-sets Ω(Z/Y) and S(Z/Y), respectively.

Recall also from §3.1 the canonical homomorphism β_{Z/Y} : 𝔖_{Z/Y} → 𝔖_{Y} which maps every permutation of a covering to the induced permutation of the base. Let 𝔗_{Z/Y} = ker β_{Z/Y}. This is the group of automorphisms over Y of the covering Z/Y, hence H^{1}(Γ, 𝔗_{Z/Y}) is in one-to-one correspondence with the set of isomorphism classes over Y of coverings of degree d of Y, where the Γ-action on Y is trivial.

The case of non-trivial Γ-action can be taken into account by twisting, see [3, §28.C]. If Z/Y is a covering of degree d, we define a non-trivial action of Γ on 𝔖_{Z/Y} by conjugation: the action of Γ on Z/Y is a group homomorphism

α: Γ → 𝔖_{Z/Y} ,

and we define, for γ ∈ Γ and f ∈ 𝔖_{Z/Y},

γ ∗ f = α(γ) ∘ f ∘ α(γ)^{−1}.

Let 𝔖′_{Z/Y} be the group 𝔖_{Z/Y} with this action of Γ, and define 𝔖′_{Y} similarly. By [3, (28.8)], there are canonical bijections

H^{1}(Γ, 𝔖′_{Z/Y}) →^{∼} H^{1}(Γ, 𝔖_{Z/Y})  and  H^{1}(Γ, 𝔖′_{Y}) →^{∼} H^{1}(Γ, 𝔖_{Y})

which map the distinguished element of H^{1}(Γ, 𝔖′_{Z/Y}), H^{1}(Γ, 𝔖′_{Y}) to the isomorphism class of the covering Z/Y and to the isomorphism class of Y, respectively. The map β_{Z/Y} is also a Γ-group homomorphism β_{Z/Y} : 𝔖′_{Z/Y} → 𝔖′_{Y}. Let 𝔗′_{Z/Y} ⊂ 𝔖′_{Z/Y} be the kernel of β_{Z/Y}. Then H^{1}(Γ, 𝔗′_{Z/Y}) is in natural one-to-one correspondence with the set C^{d}(Y) of isomorphism classes over Y of coverings of degree d of Y. (The distinguished element of H^{1}(Γ, 𝔗′_{Z/Y}) corresponds to the isomorphism class of Z/Y.)

The exact sequence of Γ-groups

1 → 𝔗′_{Z/Y} → 𝔖′_{Z/Y} ⟶^{β_{Z/Y}} 𝔖′_{Y} → 1

yields an exact sequence in cohomology

H^{0}(Γ, 𝔖′_{Y}) → H^{1}(Γ, 𝔗′_{Z/Y}) → H^{1}(Γ, 𝔖′_{Z/Y}) ⟶^{β_{Z/Y}^{1}} H^{1}(Γ, 𝔖′_{Y}).

The kernel of β_{Z/Y}^{1} is the set of isomorphism classes of coverings of degree d of Y. By [3, (28.4)], this kernel is in canonical bijection with the orbit space of H^{1}(Γ, 𝔗′_{Z/Y}) under the fixed-point group H^{0}(Γ, 𝔖′_{Y}). Note that H^{0}(Γ, 𝔖′_{Y}) is the group of permutations of Y which commute with the action of Γ; in other words, it is the group of automorphisms of the Γ-set Y,

H^{0}(Γ, 𝔖′_{Y}) = Aut_{Γ}(Y).

This group acts naturally on C^{d}(Y), and

ker β_{Z/Y}^{1} ≃ C^{d}(Y)/ Aut_{Γ}(Y).

When the Γ-action on Y is transitive, let Γ_{0} ⊂ Γ be the stabilizer of an arbitrary (but fixed) element of Y, so that Y ≃ Γ/Γ_{0}. Then we may identify 𝔗′_{Z/Y} with Map(Γ/Γ_{0}, 𝔖_{d}) and get a canonical bijection in the spirit of Shapiro’s lemma

H^{1}(Γ, 𝔗′_{Z/Y}) ≃ H^{1}(Γ_{0}, 𝔖_{d}),

see [3, (28.20)].

Whatever the action of Γ on Y, when d = 2 the group 𝔗′_{Z/Y} (≃ 𝔖_{d}^{n} where n = |Y |) is abelian, hence the set H^{1}(Γ, 𝔗′_{Z/Y}) is an abelian group. When Z/Y is the projection covering ({1, 2} × Y )/Y, the bijection H^{1}(Γ, 𝔗′_{Z/Y}) ≃ C^{2}(Y) is a group isomorphism for the group structure induced on C^{2}(Y) by the operation ∗_{Y} of §1.2. Note that this operation is generally not defined on the orbit set ker β_{Z/Y}^{1} ≃ C^{2}(Y)/ Aut_{Γ}(Y).

3.3. Torsors. As in the preceding subsection, we fix a profinite group Γ. Besides the correspondence between H^{1}(Γ, 𝔖_{n}) and the isomorphism classes of Γ-sets of n elements explained in the preceding subsection, there is also a one-to-one correspondence between H^{1}(Γ, 𝔖_{n}) and isomorphism classes of 𝔖_{n}-torsors, i.e. of Γ-sets of n! = |𝔖_{n}| elements with a free action of 𝔖_{n} (on the right) compatible with the Γ-action (on the left), see [3, (28.14)]. Combining the correspondences, we obtain a bijection between isomorphism classes of Γ-sets of n elements and 𝔖_{n}-torsors. To the isomorphism class of the Γ-set X with |X| = n corresponds the class of Σ_{n}(X), which clearly is an 𝔖_{n}-torsor. Conversely, we associate to an 𝔖_{n}-torsor Σ the class of Σ/𝔖_{n−1}. The n projections

π_{i}: Σ_{n}(X) → X, (ξ_{1}, …, ξ_{n}) ↦ ξ_{i} for i = 1, …, 2

are Γ-equivariant maps and satisfy

π_{i}((ξ_{1}, …, ξ_{n})^{σ}) = π_{σ(i)}(ξ_{1}, …, ξ_{n}) for σ ∈ 𝔖_{n}.

Definition 3.2. An 𝔖_{n}-Galois closure of a Γ-set X of n elements is a pair (Σ, π) where Σ is an 𝔖_{n}-torsor and X ⟵^{π} Σ is a covering (necessarily of degree (n − 1)!) such that π(x^{σ}) = π(x) for x ∈ Σ and σ ∈ 𝔖_{n−1}.

Every 𝔖_{n}-Galois closure of X is isomorphic to (Σ_{n}(X), π_{n}).

A similar construction can be given for coverings, since the set H^{1}(Γ, 𝔖_{d} ≀ 𝔖_{n}) classifies 𝔖_{d} ≀ 𝔖_{n}-torsors as well as coverings of degree d of Γ-sets of n elements. If Y ⟵^{π} Z is a covering of degree d of a Γ-set Y of n elements (so |Z| = nd), let Σ(Z/Y) be the set of arrays

(ζ_{ij})\begin{matrix}1≤i≤d \\ 1≤j≤n \end{matrix}

of pairwise distinct elements of Z such that π(ζ_{ij}) depends only on j for i = 1, …, n. The set Σ(Z/Y) is a 𝔖_{d} ≀ 𝔖_{n}-torsor. Its isomorphism class, viewed as an element of H^{1}(Γ, 𝔖_{d} ≀ 𝔖_{n}), corresponds to the isomorphism class of the covering Y ⟵^{π} Z. The nd projections

π_{kℓ}: Σ(Z/Y) → Z, (ζ_{ij})\begin{matrix}1≤i≤d \\ 1≤j≤n \end{matrix} ↦ ζ_{kℓ}

are Γ-equivariant maps. The projection π: Z → Y induces a 𝔖_{n}-equivariant projection

Σ(π): Σ(Z/Y) → Σ_{n}(Y), (ζ_{ij}) ↦ (π(ζ_{i1}), …, π(ζ_{in}))

and Σ_{n}(Y) ⟵^{Σ(π)} Σ(Z/Y) is a covering of degree (d!)^{n}. Moreover the diagram

Σ_{n}(Y) ⟵^{Σ(π)} Σ(Z/Y)
π_{j} ↓  ↓^{π_{ij}}
Y ⟵^{π} Z

is commutative. We say that Σ_{n}(Y) ⟵^{Σ(π)} Σ(Z/Y) is an 𝔖_{d} ≀ 𝔖_{n}-Galois closure of Y ⟵^{π} Z. (We leave it to the reader to formalize the definition of a 𝔖_{d} ≀ 𝔖_{n}-Galois closure of Y ⟵^{π} Z.)

Note that the set Ω(Z/Y) of sections of Z/Y (see §1.2) can be identified with the set Σ(Z/Y)/(𝔖_{d−1} ≀ 𝔖_{n}).

3.4. Cohomology and étale algebras. In this section, F is an arbitrary field, F_{s} is a separable closure of F and Γ = Gal(F_{s}/F) is the absolute Galois group of F. The anti-equivalence Ét_{F} ≡ Set_{Γ} induces a canonical bijection between the set of isomorphism classes of étale F -algebras of dimension n and the set of isomorphism classes of Γ-sets of n elements. Since the latter set is in one-to-one correspondence with the cohomology set H^{1}(Γ, 𝔖_{n}) (see §3.2), there is also a canonical bijection between H^{1}(Γ, 𝔖_{n}) and isomorphism classes of étale F -algebras of dimension n.

This bijection can be set up directly, by identifying 𝔖_{n} with the group of automorphisms of the split algebra F_{s}^{n}. More precisely, given an étale algebra A and an isomorphism α: F^{n} ⊗ F_{s} ⟶^{∼} A ⊗ F_{s}, the corresponding cocycle is (f_{γ})_{γ∈Γ}, where

f_{γ} = α^{−1} ∘ (1 ⊗ γ) ∘ α ∘ (1 ⊗ γ^{−1}) ∈ Aut_{F_{s}}(F^{n} ⊗ F_{s}) = 𝔖_{n}.

Conversely, given a cocycle (f_{γ})_{γ∈Γ} in 𝔖_{n}, the corresponding étale algebra is

A_{γ} = {x ∈ F_{s}^{n} | γf_{γ}(x) = x}

where Γ acts on F_{s}^{n} entrywise.

As in §3.2, the basic constructions on étale algebras of §2.1 can be interpreted in terms of morphisms of cohomology sets. Details are left to the reader, as well as the analogues for extensions of étale algebras and the cohomology of wreath products. We simply note for later use the canonical isomorphism

Quad(F) ≃ H^{1}(F, 𝔖_{2})

where Quad(F) is the group of isomorphism classes of quadratic étale F -algebras (see Proposition 2.7). For any étale F -algebra A, we also have canonical isomorphisms

Quad(A) ≃ C^{2}(X(A)) ≃ H^{1}(Γ, 𝔗′_{X(A×A)/ X(A)}),

see §3.2. The set of isomorphism classes over F of quadratic extensions of A is Quad(A)/ Aut_{F} (A). The operation ∗_{A} is generally not defined on this set.

3.5. Galois algebras. As in the preceding subsection, F is an arbitrary field and Γ is the absolute Galois group of F. Let G be a finite group. A G-Galois F -algebra is an étale F -algebra of dimension |G| with an action of G by F -algebra automorphisms such that the algebra of fixed points is F (see [3, (18.15)]). Equivalently, an étale F -algebra E of dimension |G| with an action of G is G-Galois if and only if the Γ-set X(E) is a G-torsor for the induced action of G. Therefore, the discussion of torsors in §3.3 has an analogue in terms of Galois algebras, and the set H^{1}(Γ, 𝔖_{n}) is also in one-to-one correspondence with the set of isomorphism classes of 𝔖_{n}-Galois F -algebras.

If E is an étale F -algebra of dimension n, the algebra Σ_{n}(E) has a natural action of 𝔖_{n}, for which it is an 𝔖_{n}-Galois algebra. There are n embeddings ε_{i}: E → Σ_{n}(E) corresponding to the projections π_{i}: Σ_{n}(X(E)) → X(E). They are defined explicitly as follows: for x ∈ E,

ε_{i}(x) = s_{n} · 1 ⊗ ⋯ ⊗ x ⊗ ⋯ ⊗ 1 (x in i-th position)

where s_{n} ∈ E^{⊗n} is the idempotent such that Σ_{n}(E) = s_{n}E^{⊗n}. Clearly, for σ ∈ 𝔖_{n} and x ∈ E,

ε_{σ(i)}(x) = σ(ε_{i}(x)).

Definition 3.3. An 𝔖_{n}-Galois closure of an étale F -algebra E of dimension n is a pair (σ, ε) where Σ is an 𝔖_{n}-Galois F -algebra and ε: E → Σ is an embedding such that σ(ε(x)) = ε(x) for σ ∈ 𝔖_{n} and x ∈ E.

Every 𝔖_{n}-Galois closure of E is isomorphic to (Σ_{n}(E), ε_{n}). This construction was suggested by Saltman, see [8, p. 42].

Example 3.4. Let A be a cubic étale F -algebra, i.e., dim A = 3. The choice of any of the three canonical embeddings ε_{i}: A → Σ_{3}(A) induces an isomorphism

A ⊗ Δ(A) ≃ Σ_{3}(A).

This follows from the fact that the corresponding map Σ_{3}(X) → X × Δ(X) is bijective if |X| = 3, see [3, (18.27)].

We next sketch an analogue of the Galois closure for extensions of étale algebras, on the model of the corresponding construction for coverings in §3.3.

Let B/A be an extension of degree d of an étale F -algebra A of degree n. Viewing B as an étale A-algebra of degree d, we have an 𝔖_{d}-Galois closure Σ_{d}^{A}(B) of B which is étale of degree d! over A,

Σ_{d}^{A}(B) = s_{d}^{B/A} · B^{⊗_{A}d}

where s_{d}^{B/A} is the idempotent corresponding to the characteristic function of the subset {(ξ_{1}, …, ξ_{d}) | ξ_{i} ≠ ξ_{j} for i ≠ j, and π(ξ_{i}) = π(ξ_{j}) for i = 1, …, d} in

X(B) ×_{X(A)} ⋯ ×_{X(A)} X(B) = X(B ⊗_{A} ⋯ ⊗_{A} B).

There are d canonical embeddings ε_{i}^{A}: B → Σ_{d}^{A}(B) and a canonical embedding j: A → Σ_{d}^{A}(B) corresponding to the A-algebra structure on Σ_{d}^{A}(B). Define

Σ(B/A) = j^{⊗n}(s_{n}^{A}) · Σ_{d}^{A}(B)^{⊗n}.

As for Proposition 2.8 we have

Proposition 3.5. X(Σ(B/A)) = Σ(X(B)/ X(A)).

The algebra Σ(B/A) is an extension of Σ_{n}(A) of degree (d!)^{n} and there exist nd canonical embeddings

(5)  ε_{ij}: B → Σ(B/A), 1 ≤ i ≤ d, 1 ≤ j ≤ n

such that the diagram

Σ_{n}(A) ⟶ Σ(B/A)
ε_{i}↑  ↑^{ε_{ij}}
A ⟶ B

is commutative for all i and j. We say that the extension Σ(B/A)/Σ_{n}(A) is an 𝔖_{d} ≀ 𝔖_{n}-Galois closure of the extension B/A. Since Σ_{d}^{A}(B)^{𝔖_{d−1}} = B, we have

Σ(B/A)^{𝔖_{d−1}≀𝔖_{n}} = Ω(B/A).

If d = 2, each of the canonical embeddings ε_{1}^{A}, ε_{2}^{A}: B → Σ_{2}^{A}(B) is an isomorphism (and ε_{2}^{A} = ε_{1}^{A} ∘ γ_{B/A}), and j: A → Σ_{2}^{A}(B) = B is the inclusion. The algebra

Σ(B/A) = j^{⊗n}(s_{n}^{A}) · B^{⊗n}

is an extension of degree 2^{n} of Σ_{n}(A), and

Ω(B/A) ≃ Σ(B/A)^{𝔖_{n}}.

Example 3.6. Suppose, as in Example 2.9, that A and B are split of dimensions 3 and 6 respectively, with minimal idempotents e_{1}, e_{2}, e_{3} and f_{1}, f′_{1}, f_{2}, f′_{2}, f_{3}, f′_{3} such that

e_{i} = f_{i} + f′_{i} for i = 1, 2, 3.

The algebra Σ(B/A) is split. Its 48 minimal idempotents are

\begin{matrix}f_{σ(1)} ⊗ f_{σ(2)} ⊗ f_{σ(3)}, & f′_{σ(1)} ⊗ f′_{σ(2)} ⊗ f′_{σ(3)}, \\ f_{σ(1)} ⊗ f′_{σ(2)} ⊗ f′_{σ(3)}, & f′_{σ(1)} ⊗ f_{σ(2)} ⊗ f_{σ(3)}, \\ f′_{σ(1)} ⊗ f_{σ(2)} ⊗ f′_{σ(3)}, & f_{σ(1)} ⊗ f′_{σ(2)} ⊗ f_{σ(3)}, \\ f′_{σ(1)} ⊗ f′_{σ(2)} ⊗ f_{σ(3)}, & f_{σ(1)} ⊗ f_{σ(2)} ⊗ f′_{σ(3)}, \end{matrix}

where σ varies in 𝔖_{3}. The action of 𝔖_{3} on these idempotents is clear, and the fixed subalgebra is Ω(B/A) as described in Example 2.9.

Proposition 3.7. Let B/A be an extension of étale algebras of degree d and let n = dim_{F} A. If b ∈ B is a generator of B as F -algebra, then the nd elements ε_{ij}(b) generate Σ(B/A) over Σ_{n}(A).

Proof. Let Σ′ be the subalgebra of Σ(B/A) generated over Σ(A) by the ε_{ij}(b). We show that Σ′ = Σ(B/A). We may assume that A and B are split and we assume for simplicity that n = 3 and d = 2. We use the notations of Example 2.9. Let

b = β_{1}f_{1} + β′_{1}f′_{1} + β_{2}f_{2} + β′_{2}f′_{2} + β_{3}f_{3} + β′_{3}f′_{3}

with β_{1}, …, β′_{3} ∈ F, hence

\overline{b} = β′_{1}f_{1} + β_{1}f′_{1} + β′_{2}f_{2} + β_{2}f′_{2} + β′_{3}f_{3} + β_{3}f′_{3}.

Since b generates B, the 6 elements β_{i}, β′_{j} are pairwise different. We have

ε_{11}(b) = s_{3}^{A} · b ⊗ 1 ⊗ 1 and ε_{21}(b) = s_{3}^{A} · \overline{b} ⊗ 1 ⊗ 1.

Thus

\begin{aligned}ε_{11}(b)(e_{1} ⊗ e_{2} ⊗ e_{3}) & = & (β_{1}f_{1} + β′_{1}f′_{1}) ⊗ e_{2} ⊗ e_{3} and \\ ε_{21}(b)(e_{1} ⊗ e_{2} ⊗ e_{3}) & = & (β′_{1}f_{1} + β_{1}f′_{1}) ⊗ e_{2} ⊗ e_{3} \end{aligned}

are elements of Σ′. It follows that f_{1} ⊗ e_{2} ⊗ e_{3} and f′_{1} ⊗ e_{2} ⊗ e_{3} are in Σ′. Hence all the minimal idempotents of Σ(B/A) are in Σ′ and Σ′ = Σ(B/A). □

4. The symmetric group on four elements

In the rest of this paper, we focus on various aspects of étale algebras of dimension 4 (called quartic étale algebras) which, as explained in the preceding sections, can be viewed from the perspective of Γ-sets of 4 elements, or of the cohomology of 𝔖_{4}, or of 𝔖_{4}-torsors, or of 𝔖_{4}-Galois algebras. It turns out that there is a group isomorphism

𝔖_{2} × 𝔖_{4} ≃ 𝔖_{2} ≀ 𝔖_{3}

which relates the various “quartic” notions listed above to those associated with the cohomology of 𝔖_{2} ≀ 𝔖_{3}: quadratic extensions of cubic étale algebras, double coverings of sets of 3 elements, 𝔖_{2} ≀ 𝔖_{3}-torsors and 𝔖_{2} ≀ 𝔖_{3}-Galois algebras. We explain this relation in the simplest case, namely Γ-sets and coverings, and then give the cohomological viewpoint in the next subsection. In the last two subsections, we give explicit constructions of R(Q) for a quartic algebra Q, making clear that this algebra is related to the resolvent cubic of quartic equations, and of Ω(B/A) and S(B/A) for a quadratic extension of a cubic algebra A.

4.1. Sets of four elements and double coverings. In this subsection, Γ is an arbitrary profinite group. Suppose X is a Γ-set with |X| = 4, as in Example 1.1, where the constructions of Λ_{2}(X) and R(X) are made explicit. Our first observation concerns the discriminants of Λ_{2}(X) and R(X):

Proposition 4.1. The map which carries (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) ∈ Σ_{4}(X) to

({{ξ_{1}, ξ_{2}}, {ξ_{3}, ξ_{4}}}, {{ξ_{1}, ξ_{3}}, {ξ_{2}, ξ_{4}}}, {{ξ_{1}, ξ_{4}}, {ξ_{2}, ξ_{3}}}) ∈ Σ_{3}(R(X))

induces a canonical isomorphism of Γ-sets

Δ(X) →^{∼} Δ(R(X)).

Moreover, the Γ-action on Δ(Λ_{2}(X)) is trivial.

The proof is a straightforward verification. To see that the Γ-action on Δ(Λ_{2}(X)) is trivial, it suffices to observe that every transposition on X —hence every permutation of X— induces an even permutation of Λ_{2}(X). For another approach, see Proposition 4.7.

To get a better grasp of the various constructions associated with X, it is useful to think of X as the set of diagonals of a cube.^{1} Each pair of diagonals determines a diagonal plane (passing through an edge and its opposite), hence Λ_{2}(X) is identified with the set of diagonal planes of the cube. The map γ_{X} carries each diagonal plane to the plane through parallel edges, and R(X) can therefore be identified with the set of directions of the edges. The canonical map R(X) ⟵^{ε} Λ_{2}(X) maps each diagonal plane to the direction of the edges it contains. The set Ω(Λ_{2}(X)/ R(X)) consists of (unordered) triples of diagonal planes with different edge directions. For each such triple τ, either the intersection of the planes is a diagonal, or the intersection is just the center of the cube. However, if the intersection is a diagonal, then the intersection of the complementary triple \overline{τ} = γ_{Λ_{2}(X)/ R(X)}(τ) is the center. Therefore, we may associate to the pair \{τ, \overline{τ}\} ∈ S(Λ_{2}(X)/ R(X)) a unique diagonal in X, and obtain a map

Φ: S(Λ_{2}(X)/ R(X)) → X.

Proposition 4.2. For |X| = 4, the map

Φ: S(Λ_{2}(X)/ R(X)) → X

is a canonical isomorphism of Γ-sets.

Proof. From the definition, it is clear that Φ is Γ-equivariant. Bijectivity of Φ is checked by direct inspection. □

To put this result into perspective, consider the full subcategory Set_{Γ}^{4} of Set_{Γ} whose objects are the Γ-sets of four elements, and the category Cov_{Γ}^{2≀3} of double coverings of Γ-sets of three elements, with morphisms of coverings. There are functors

Λ: Set_{Γ}^{4} → Cov_{Γ}^{2≀3} and S: Cov_{Γ}^{2≀3} → Set_{Γ}^{4}

defined by

Λ(X) = Λ_{2}(X)/ R(X)  and  S(Y ⟵^{π} Z) = S(Z/Y).

Proposition 4.2 yields a natural equivalence between S ∘ Λ and the identity on Set_{Γ}^{4}.

1We are indebted to F. Buekenhout for his suggestion to use geometric language in this context.

To investigate the composition Λ ∘ S, suppose Y ⟵^{π} Z is a double covering of a Γ-set Y with |Y | = 3. (See Example 1.3 for an explicit description of Ω(Z/Y) and S(Z/Y).) We may consider Z as the set of faces of a cube, Y as the set of directions of edges, and π as the map which carries each face to the orthogonal direction. Then Ω(Z/Y) is the set of (unordered) triples of faces which are not pairwise parallel. Since the faces in each such triple meet at one vertex, we may view Ω(Z/Y) as the set of vertices of the cube. The map γ_{Z/Y} carries each vertex to its opposite, hence S(Z/Y) is the set of diagonals of the cube. As in the discussion before Proposition 4.2, we may then identify Λ_{2}(S(Z/Y)) with the set of diagonal planes and R(S(Z/Y)) with the set of edge directions. It is then clear that R(S(Z/Y)) is canonically identified with Y, but there is no canonical identification of Λ_{2}(S(Z/Y)) with Z.

As we now show, we may however define a canonical bijection

Δ(Z) ∗ Λ_{2}(S(Z/Y)) →^{∼} Z,

hence an isomorphism of coverings between Y ⟵^{π} Z and the covering R(S(Z/Y)) ⟵^{ε} Δ(Z)∗Λ_{2}(S(Z/Y)) induced by the canonical covering R(S(Z/Y)) ⟵^{ε} Λ_{2}(S(Z/Y)). (We denote both coverings by ε.)

Our first goal is to give a geometrical interpretation of the set Δ(Z). Recall the map

δ_{Z}: Ω(Z/Y) → Δ(Z)

of (2). By Proposition 1.4, this map is onto. It may therefore be used to consider Δ(Z) as a quotient of Ω(Z/Y), the set of vertices of the cube. It is easily checked that the four vertices which have the same image under δ_{Z} are the vertices of a regular tetrahedron whose edges are the diagonals of the faces of the cube. Therefore, we may identify Δ(Z) with the set {T_{1}, T_{2}} of such tetrahedra. Given a diagonal plane λ ∈ Λ_{2}(S(Z/Y)) and a tetrahedron T ∈ Δ(Z), there is a unique face z ∈ Z whose intersection with λ is an edge of T. The same face z intersects the “complementary” plane \overline{λ} following an edge of the “complementary” tetrahedron \overline{T}. Therefore, the map (T, λ) ↦ z induces a well-defined map

Ψ: Δ(Z) ∗ Λ_{2}(S(Z/Y)) → Z.

Proposition 4.3. The map Ψ defines an isomorphism of coverings between

R(S(Z/Y)) ⟵^{ε} Δ(Z) ∗ Λ_{2}(S(Z/Y))  and  Y ⟵^{π} Z.

Proof. The map Ψ is clearly equivariant. The other properties are checked by direct inspection. □

This proposition shows that Λ ∘ S is not equivalent to the identity. However, when Δ(Z) is a trivial Γ-set the proposition yields an isomorphism between Z/Y and Λ ∘ S(Z/Y):

Corollary 4.4. If the Γ-action on Δ(Z) is trivial, then there is an isomorphism of coverings between

R(S(Z/Y)) ⟵^{ε} Λ_{2}(S(Z/Y))  and  Y ⟵^{π} Z.

Proof. This readily follows from Proposition 4.3 and Proposition 1.5(b). □

Corollary 4.4 applies in particular to double coverings of the form Λ_{2}(X)/ R(X), for X a Γ-set with |X| = 4, by Proposition 4.1. Therefore, Λ ∘ S(X) ≃ X.

Theorem 4.5. The functors Λ and S define a canonical one-to-one correspondence between the set of isomorphism classes of Γ-sets of 4 elements and the set of isomorphism classes of double coverings Z/Y of Γ-sets Y of 3 elements with trivial action on Δ(Z).

Remark. ^{2} For X, X′ ∈ Set_{Γ}^{4}, every morphism of coverings f: Λ_{2}(X)/ R(X) → Λ_{2}(X′)/ R(X′) induces a morphism S(Λ_{2}(X)/ R(X)) → S(Λ_{2}(X′)/ R(X′)), hence, by Proposition 4.2, a morphism f̃: X → X′. The functor Λ carries f̃ to f, hence it is full. Since S ∘ Λ is equivalent to the identity, the functor Λ is also faithful. Moreover, Corollary 4.4 shows that every covering Z/Y ∈ Cov_{Γ}^{2≀3} such that the Γaction on Δ(Z) is trivial is isomorphic to a covering of the form Λ(X). Therefore, it follows from [6, Theorem 1, p. 93] that Λ is an equivalence of categories from Set_{Γ}^{4} to the full subcategory of Cov_{Γ}^{2≀3} whose objects are the coverings Z/Y with trivial Γ-action on Δ(Z).

In order to take into account the double coverings of Γ-sets of three elements which have non-trivial action on the discriminant, we consider the product category Set_{Γ}^{2} × Set_{Γ}^{4} whose objects are pairs (U, X) of Γ-sets with |U| = 2 and |X| = 4, and extend Λ and S to functors

Λ̂: Set_{Γ}^{2} × Set_{Γ}^{4} → Cov_{Γ}^{2≀3}  and  Ŝ: Cov_{Γ}^{2≀3} → Set_{Γ}^{2} × Set_{Γ}^{4}

defined by

Λ̂(U, X) = (U ∗ Λ_{2}(X))/ R(X)  and  Ŝ(Y ⟵^{π} Z) = (Δ(Z), S(Z/Y)).

Proposition 4.3 yields a natural equivalence between Λ̂ ∘ Ŝ and the identity on Cov_{Γ}^{2≀3}.

On the other hand, for U, X with |U| = 2 and |X| = 4, we have canonical isomorphisms

S(U ∗ Λ_{2}(X)/ R(X)) ≃ S(Λ_{2}(X)/ R(X)) ≃ X

by Propositions 1.8 and 4.2, and

(6)  Δ(U ∗ Λ_{2}(X)) ≃ Δ(U × R(X)) ∗ Δ(Λ_{2}(X)) ≃ U ∗ Δ(Λ_{2}(X)),

by Propositions 1.6 and 1.7. The Γ-action on Δ(Λ_{2}(X)) is trivial by Proposition 4.1, hence the right-most Γ-set in (6) is isomorphic to U by Proposition 1.2(b). Note that the latter isomorphism is not canonical, hence Ŝ∘ Λ̂ is not naturally equivalent to the identity on Set_{Γ}^{2} × Set_{Γ}^{4}. However, since Ŝ∘ Λ̂(U, X) ≃ (U, X), we have an isomorphism between sets of isomorphism classes.

Theorem 4.6. The functors Λ̂ and Ŝ define a canonical one-to-one correspondence between the set of isomorphism classes of pairs of Γ-sets (U, X) with |U| = 2 and |X| = 4 and the set of isomorphism classes of double coverings of Γ-sets with three elements.

An alternative proof in cohomology can be derived from Diagram (10). Theorems 4.5 and 4.6 have analogues in terms of quartic étale algebras and double coverings of cubic algebras, whose statements are left to the reader.

2The authors are indebted to F. Borceux for enlightening comments about this remark.

Remark. The functor Ŝ is faithful since Λ̂ ∘ Ŝ is equivalent to the identity. Moreover, every (U, X) ∈ Set_{Γ}^{2} × Set_{Γ}^{4} is isomorphic to an object of the form Ŝ(Z/Y) (namely, Z/Y = Λ̂(U, X)). Furthermore, every morphism f: Ŝ(Z/Y) → Ŝ(Z′/Y′) induces a morphism Λ̂(f): Λ̂ ∘ Ŝ(Z/Y) → Λ̂ ∘ Ŝ(Z′/Y′), hence by Proposition 4.3 a morphism f̃: Z/Y → Z′/Y′. We may check that f = Ŝ(f̃), hence the functor Ŝ is full. By [6, Theorem 1, p. 93], it defines an equivalence of categories Cov_{Γ}^{2≀3} ≡ Set_{Γ}^{2} × Set_{Γ}^{4}.

4.2. Cohomology. This subsection presents the cohomological perspective on Theorem 4.6. We use the same notation as in the preceding subsection.

Let U = {1, 2} and X = {1, 2, 3, 4} with trivial Γ-action. The group of automorphisms of (U, X) in the category Set_{Γ}^{2} × Set_{Γ}^{4} is 𝔖_{2} × 𝔖_{4}, and since Ŝ∘ Λ̂(U, X) ≃ (U, X), the functor Λ̂ yields an isomorphism

(7)  λ̂: 𝔖_{2} × 𝔖_{4} →^{∼} Aut(U ∗ Λ_{2}(X)/ R(X)) ≃ 𝔖_{2} ≀ 𝔖_{3}.

For definiteness, consider 𝔖_{2} ≀ 𝔖_{3} as the group of automorphisms of the covering

Y = {1, 2, 3} ⟵^{π_{2}} {1, 2} × {1, 2, 3} = Z,

where Γ acts trivially on Y and Z. The right isomorphism in (7) depends on the choice of an isomorphism U ∗ Λ_{2}(X)/ R(X) ≃ Z/Y.

Similarly, the functor Ŝ yields an isomorphism

(8)  ŝ: 𝔖_{2} ≀ 𝔖_{3} →^{∼} Aut(Δ(Z), S(Z/Y)) ≃ 𝔖_{2} × 𝔖_{4}

where again the latter isomorphism is given by identifications Δ(Z) ≃ {1, 2} and S(Z/Y) ≃ {1, 2, 3, 4}. The isomorphisms λ̂ and ŝ induce bijections

H^{1}(Γ, 𝔖_{2} × 𝔖_{4}) ≃ H^{1}(Γ, 𝔖_{2} ≀ 𝔖_{3}).

Since these cohomology sets are in one-to-one correspondence with the sets of isomorphism classes in Set_{Γ}^{2} × Set_{Γ}^{4} and Cov_{Γ}^{2≀3} respectively (see §3.2), we thus recover Theorem 4.6.

The isomorphisms λ̂ and ŝ can also be described in purely group-theoretical terms. The subgroup 𝔖_{2} = 𝔖_{2} × {1} ⊂ 𝔖_{2} × 𝔖_{4} is mapped to the “diagonal” subgroup 𝔖_{2} ⊂ 𝔖_{2} ≀ 𝔖_{3}, which is the center of 𝔖_{2} ≀ 𝔖_{3}. On the other hand, the restriction of λ̂ to 𝔖_{4} = {1} × 𝔖_{4} is a homomorphism

λ: 𝔖_{4} → 𝔖_{2} ≀ 𝔖_{3}

which may be described as the action of 𝔖_{4} by conjugation on its transpositions. Indeed, 𝔖_{4} contains six transpositions, which sit by pairs in the three Sylow 2subgroups of 𝔖_{4}. The map which carries each transposition to the unique Sylow 2-subgroup which contains it is a double covering of a set of three elements. Note that the composition of λ and the canonical homomorphism β: 𝔖_{2} ≀ 𝔖_{3} → 𝔖_{3} is the surjective homomorphism

ρ: 𝔖_{4} → 𝔖_{3}

which is the action of 𝔖_{4} on its three Sylow 2-subgroups. (Alternately, the map ρ may be identified with the canonical homomorphism 𝔖_{X} → 𝔖_{R(X)} for X = {1, 2, 3, 4}, since there is a canonical one-to-one correspondence between R(X) and the Sylow 2-subgroups of 𝔖_{X}.) The kernel of ρ is the Vierergruppe 𝔙.

By definition, it is clear that the first component of ŝ is the signature map

sgn: 𝔖_{2} ≀ 𝔖_{3} ⊂ 𝔖_{6} → 𝔖_{2},

since the map 𝔖_{Z/Y} → 𝔖_{Δ(Z)} is the signature. The second component is a homomorphism

s: 𝔖_{2} ≀ 𝔖_{3} → 𝔖_{4}

which is the action of 𝔖_{2} ≀ 𝔖_{3} on its four Sylow 3-subgroups. (There is a natural one-to-one correspondence between the Sylow 3-subgroups of 𝔖_{Z/Y} and S(Z/Y).) The image of λ is the kernel of sgn, by Proposition 4.1 or by Proposition 3.1, and the map s splits λ (if the Sylow 3-subgroups of 𝔖_{2} ≀ 𝔖_{3} are suitably indexed). The maps ρ, λ and β, and the inclusions ι, η, are part of the following commutative diagram with exact rows and columns:

(9)  \begin{aligned}& & 1 & & 1 \\ & & ↓ & & ↓ \\ 1 & ⟶ & 𝔙 & ⟶^{ι} & 𝔖_{4} & ⟶^{ρ} & 𝔖_{3} & ⟶ & 1 \\ & & ↓ & & ↓^{λ} & & ∥ \\ 1 & ⟶ & 𝔖_{2}^{3} & ⟶^{η} & 𝔖_{2} ≀ 𝔖_{3} & ⟶^{β} & 𝔖_{3} & ⟶ & 1 \\ & & ^{σ}↓ & & ↓^{sgn} \\ & & 𝔖_{2} & & 𝔖_{2} \\ & & ↓ & & ↓ \\ & & 1 & & 1 \end{aligned}

where σ is the sum. Since the exact sequences in this diagram are split, there is a corresponding commutative diagram of exact sequences in cohomology:

(10)

1  1
↓  ↓
1 ⟶ H^{1}(Γ, 𝔙) ⟶^{ι^{1}} H^{1}(Γ, 𝔖_{4}) ⟶^{ρ^{1}} H^{1}(Γ, 𝔖_{3}) ⟶ 1
↓  ↓_{λ^{1}}  ∥
1 ⟶ H^{1}(Γ, 𝔖_{2}^{3}) ⟶^{η^{1}} H^{1}(Γ, 𝔖_{2} ≀ 𝔖_{3}) ⟶^{β^{1}} H^{1}(Γ, 𝔖_{3}) ⟶ 1
σ^{1}↓  ↓^{sgn}^{1}
H^{1}(Γ, 𝔖_{2})  H^{1}(Γ, 𝔖_{2})
↓  ↓
1  1.

Cohomology yields an alternative proof of Proposition 4.1:

Proposition 4.7. For any Γ-set X with |X| = 4,

Δ(X) ≃ Δ(R(X)).

Moreover, the Γ-action on Δ(Λ_{2}(X)) is trivial.

Proof. The commutative diagram

𝔖_{4} ⟶^{ρ} 𝔖_{3}
sgn↓  ↓^{sgn}
𝔖_{2}  𝔖_{2}

induces a commutative diagram in cohomology

H^{1}(Γ, 𝔖_{4}) ⟶^{ρ^{1}} H^{1}(Γ, 𝔖_{3})
sgn^{1}↓  ↓^{sgn}^{1}
H^{1}(Γ, 𝔖_{2})  H^{1}(Γ, 𝔖_{2}).

The first part of the proposition follows since ρ^{1} maps the isomorphism class of X to the isomorphism class of R(X), and sgn^{1} maps the isomorphism class of any Γ-set to the isomorphism class of its discriminant. The second part follows from the fact that

H^{1}(Γ, 𝔖_{4}) ⟶^{λ^{1}} H^{1}(Γ, 𝔖_{2} ≀ 𝔖_{3}) ⟶^{sgn^{1}} H^{1}(Γ, 𝔖_{2})

is a zero-sequence. □

As another application of cohomology, we describe the quartic étale algebras which have a given resolvent cubic.

Let R be a Γ-set of three elements, and let X_{0} = R ∐ {0} be the Γ-set of four elements obtained by adjoining to R a fixed point 0. To each partition of X_{0} into 2-element subsets, we may associate the unique element r ∈ R such that {0, r} is in the partition, and thus identify

R(X_{0}) = R.

As in §3.2, we let Γ act by conjugation on the groups 𝔖_{X_{0}}, 𝔖_{R}, and denote by 𝔖′_{X_{0}}, 𝔖′_{R} the Γ-groups thus defined. The inclusion R ↪ X_{0} yields a Γ-equivariant embedding 𝔖′_{R} ↪ 𝔖′_{X_{0}} which splits the map ρ: 𝔖′_{X_{0}} → 𝔖′_{R}. The split exact sequence

1 → 𝔙′_{X_{0}} ⟶^{ι} 𝔖′_{X_{0}} ⟶^{ρ} 𝔖′_{R} → 1

yields an exact sequence in cohomology

(11)  1 → H^{1}(Γ, 𝔙′_{X_{0}}) ⟶^{ι^{1}} H^{1}(Γ, 𝔖′_{X_{0}}) ⟶^{ρ^{1}} H^{1}(Γ, 𝔖′_{R}) → 1,

and the isomorphism classes of X ∈ Set_{Γ}^{4} such that R(X) ≃ R are in one-to-one correspondence with ker ρ^{1} = im ι^{1}. They form a pointed set with the isomorphism class of X_{0} as distinguished element. Note that exactness of the sequence (11) does not mean that ι^{1} is injective. In fact, the group Aut_{Γ}(R) = H^{0}(Γ, 𝔖′_{R}) acts on H^{1}(Γ, 𝔙′_{X_{0}}), and im ι^{1} is in canonical bijection with the orbit set H^{1}(Γ, 𝔙′_{X_{0}})/ Aut_{Γ}(R), by [3, (28.4)].

To give a more explicit description, we use a variant of Diagram (10). First, observe that we may identify Λ_{2}(X_{0}) to {1, 2} × R, as follows: we map a 2-element subset U ⊂ X_{0} to (1, r) if 0 ∉ U and r ∉ U, to (2, r) if U = {0, r}. We may then identify the double covering Λ_{2}(X_{0})/ R(X_{0}) to

R ⟵^{π_{2}} {1, 2} × R.

Let Z_{0} = {1, 2} × R. As above, we let Γ act by conjugation on 𝔖_{Z_{0}/R}, and denote by 𝔖′_{Z_{0}/R} the corresponding Γ-group. As in §3.2, let 𝔗′_{Z_{0}/R} be the kernel of the canonical map β_{Z_{0}/R}: 𝔖′_{Z_{0}/R} → 𝔖′_{R}. The exact sequence

1 → 𝔗′_{Z_{0}/R} ⟶^{η} 𝔖′_{Z_{0}/R} ⟶^{β_{Z_{0}/R}} 𝔖′_{R} → 1

is split, and induces an exact sequence in cohomology

1 → H^{1}(Γ, 𝔗′_{Z_{0}/R}) ⟶^{η^{1}} H^{1}(Γ, 𝔖′_{Z_{0}/R}) ⟶^{β^{1}} H^{1}(Γ, 𝔖′_{R}) → 1.

As above, there is a canonical bijection

im η^{1} = ker β^{1} ≃ H^{1}(Γ, 𝔗′_{Z_{0}/R})/ Aut_{Γ}(R).

The orbit set on the right side may therefore be identified with the set of isomorphism classes of double coverings of R, see §3.2. Consider the commutative diagram analogous to (10),

(12)

1  1
↓  ↓
H^{1}(Γ, 𝔙′_{X_{0}}) ⟶^{ι^{1}} H^{1}(Γ, 𝔖′_{X_{0}}) ⟶^{ρ^{1}} H^{1}(Γ, 𝔖′_{R})
↓  ↓_{λ^{1}}  ∥
H^{1}(Γ, 𝔗′_{Z_{0}/R}) ⟶^{η^{1}} H^{1}(Γ, 𝔖′_{Z_{0}/R}) ⟶^{β^{1}} H^{1}(Γ, 𝔖′_{R})
σ^{1}↓  ↓^{sgn}^{1}
H^{1}(Γ, 𝔖_{2})  H^{1}(Γ, 𝔖_{2})
↓  ↓
1  1.

The left vertical sequence is an exact sequence of groups. It shows that H^{1}(Γ, 𝔙′_{X_{0}}) can be identified with the kernel of σ^{1}. Recall from §3.2 that H^{1}(Γ, 𝔗′_{Z_{0}/R}) is in canonical bijection with the set C^{2}(R) of isomorphism classes over R of double coverings of R, and that H^{1}(Γ, 𝔖_{2}) classifies Γ-sets of two elements up to isomorphism. By commutativity of the lower square in (12), the map σ^{1} carries every double covering to the isomorphism class of its discriminant. Therefore, we may identify H^{1}(Γ, 𝔙′_{X_{0}}) with the group C_{0}^{2}(R) of isomorphism classes over R of double coverings of R with trivial discriminant,

H^{1}(Γ, 𝔙′_{X_{0}}) = C_{0}^{2}(R).

We have thus shown:

Proposition 4.8. The set of isomorphism classes of sets X of four elements such that R(X) ≃ R is in canonical bijection with the set C_{0}^{2}(R)/ Aut_{Γ}(R) of isomorphism classes of double coverings of R with trivial discriminant.

Suppose now Γ is the absolute Galois group of a field F with separable closure F_{s}, and let A be a cubic étale F -algebra. Using the anti-equivalence Set_{Γ} ≡ Ét_{F}, we may translate Proposition 4.8 into the following statement, where we denote by Quad_{0}(A) the set of isomorphism classes over A of quadratic extensions of A whose discriminant (as F -algebra) is trivial:

Proposition 4.9. The set of isomorphism classes of quartic étale F -algebras Q with R(Q) ≃ A is in canonical bijection with the set Quad_{0}(A)/ Aut_{F} (A) of F - isomorphism classes of quadratic extensions of A with trivial discriminant.

The next proposition gives an explicit description of the group Quad_{0}(A).

Proposition 4.10. Let N^{1}(A) be the (multiplicative) group of elements of A of norm 1 and let T^{0}(A) be the (additive) group of elements of A of trace 0.

(a) If char F ≠ 2, Quad_{0}(A) ≃ N^{1}(A)/N^{1}(A)^{2}.

(b) If char F = 2, Quad_{0}(A) ≃ T^{0}(A)/℘(T^{0}(A)), where ℘ is the Artin-Schreier

map ℘(x) = x^{2} − x.

Proof. If A is a field, the action of Γ on X(A) is transitive. Letting Γ_{0} ⊂ Γ be the absolute Galois group of a copy of A in F_{s}, we have Quad(A) ≃ H^{1}(Γ_{0}, 𝔖_{2}) as observed in §3.2, and the map σ^{1} can be interpreted as the corestriction

H^{1}(Γ_{0}, 𝔖_{2}) → H^{1}(Γ, 𝔖_{2}).

If char F ≠ 2, we identify 𝔖_{2} with {1, −1} ⊂ F_{s}^{×}. The exact sequence

1 → 𝔖_{2} → F_{s}^{×} ⟶^{2} F_{s}^{×} → 1

yields isomorphisms

H^{1}(Γ, 𝔖_{2}) ≃ F^{×}/F^{×2}  and  H^{1}(Γ_{0}, 𝔖_{2}) ≃ A^{×}/A^{×2}

under which the corestriction corresponds to a map induced by the norm. Its kernel is N^{1}(A)/N^{1}(A)^{2} since if y ∈ A^{×} is such that N_{A/F} (y) = z^{2} ∈ F^{×2}, then N_{A/F} (y^{3}z^{−2}) = 1.

If char F = 2, we identify 𝔖_{2} with {0, 1} ⊂ F_{s}. The exact sequence

0 → 𝔖_{2} → F_{s} ⟶^{℘} F_{s} → 0

yields isomorphisms

H^{1}(Γ, 𝔖_{2}) ≃ F/℘(F)  and  H^{1}(Γ_{0}, 𝔖_{2}) ≃ A/℘(A)

under which the corestriction corresponds to a map induced by the trace. Its kernel is T^{0}(A)/℘(T^{0}(A)) since if T_{A/F} (y) = z^{2} − z, then T_{A/F} (y − z^{2} + z) = 0.

If A is not a field, it decomposes into a direct product of fields,

A ≃ F × K  or  A ≃ F × F × F.

In the first case,

Quad(A) ≃ Quad(F) × Quad(K) ≃ \begin{cases}(F^{×}/F^{×2}) × (K^{×}/K^{×2}) & if char F ≠ 2, \\ (F/℘(F)) × (K/℘(K)) & if char F = 2, \end{cases}

and the map σ^{1} can be again interpreted as induced by the norm or the trace. We may then use the same arguments as above. The case where A ≃ F × F × F is left to the reader. □

Following §3.5, the set H^{1}(Γ, 𝔖′_{R}) for R = X(A) is also in one-to-one correspondence with the set of isomorphism classes of 𝔖_{3}-Galois F -algebras, where the distinguished element corresponds to the isomorphism class of the 𝔖_{3}-Galois closure Σ_{3}(A). Likewise, the set H^{1}(Γ, 𝔖′_{X_{0}}) classifies 𝔖_{4}-Galois F -algebras up to isomorphism, with the class of Σ_{4}(F × A) as distinguished element. The upper exact sequence of Diagram (12) shows that the 𝔖_{4}-Galois F -algebras M which are the 𝔖_{4}-Galois closure of an étale quartic F -algebra Q with R(Q) ≃ A are in oneto-one correspondence with Quad_{0}(A)/ Aut_{F} (A). Using Proposition 4.10, we may make this correspondence explicit as follows:

Proposition 4.11. Let A be a cubic étale F -algebra, identified with a subalgebra of its 𝔖_{3}-Galois closure Σ_{3}(A), and let ρ ∈ 𝔖_{3} be an element of order 3.

(a) If char F ≠ 2, let a ∈ A^{×} be such that N_{A/F} (a) = 1, and set

M = Σ_{3}(A)[\sqrt{a}, \sqrt{ρ(a)}, \sqrt{ρ^{2}(a)}].

(b) If char F = 2, let a ∈ A be such that T_{A/F} (a) = 0, and set

M = Σ_{3}(A)[℘^{−1}(a), ℘^{−1}(ρ(a)), ℘^{−1}(ρ^{2}(a))].

In each case, there is a 𝔖_{4}-action on M which endows it with the structure of an 𝔖_{4}-Galois algebra. The quartic subalgebra Q = M^{𝔖_{3}} satisfies R(Q) ≃ A. Moreover, every 𝔖_{4}-Galois F -algebra which is the 𝔖_{4}-Galois closure of a quartic étale F -algebra Q with R(Q) ≃ A is of this form.

Remark. Similar constructions are described by Serre (see [10]^{3}) and by Weil (for the construction of dyadic field extensions with Galois group 𝔖_{4}, see [13, Section 31]).

4.3. Quartic étale algebras. In this subsection, our goal is to make explicit the relation between resolvent cubics of quartic polynomials and the construction of R(Q) for Q a quartic étale F -algebra. Our first observation is a direct consequence of Proposition 4.1 (see also Proposition 4.7).

Proposition 4.12. Let Q be a quartic étale F -algebra. There is a canonical isomorphism

Δ(R(Q)) →^{∼} Δ(Q).

Moreover, Δ(Λ_{2}(Q)) ≃ F × F.

Proof. The proposition readily follows from Proposition 4.1 under the anti-equi-valence Ét_{F} ≡ Set_{Γ}, since the Δ, R and Λ_{2} construction commute with the functor X. □

Recall from [12] that “the” resolvent cubic of a quartic polynomial

(13)  f(u) = u^{4} − α_{1}u^{3} + α_{2}u^{2} − α_{3}u + α_{4}

with roots u_{1}, u_{2}, u_{3}, u_{4} in an algebraic closure, is the polynomial g(v) with roots

v_{1} = (u_{1} + u_{2})(u_{3} + u_{4}), v_{2} = (u_{1} + u_{3})(u_{2} + u_{4}), v_{3} = (u_{1} + u_{4})(u_{2} + u_{3}).

This polynomial has the form

(14)  g(v) = v^{3} − β_{1}v^{2} + β_{2}v − β_{3}

where

\begin{aligned}β_{1} & = 2α_{2}, \\ β_{2} & = α_{1}α_{3} + α_{2}^{2} − 4α_{4}, \\ β_{3} & = α_{1}α_{2}α_{3} − α_{1}^{2}α_{4} − α_{3}^{2}. \end{aligned} ; (15)

3We are indebted to J-P. Serre for calling our attention to this reference.

An alternative resolvent cubic suggested by Lagrange [4, (32), p. 266] in characteristic different from 2 has roots

w_{1} = (u_{1}+u_{2}−u_{3}−u_{4})^{2},  w_{2} = (u_{1}−u_{2}+u_{3}−u_{4})^{2},  w_{3} = (u_{1}−u_{2}−u_{3}+u_{4})^{2}.

Since w_{i} = α_{1}^{2} − 4v_{i} for i = 1, 2, 3, this polynomial has the form

(16)  h(w) = −4^{3}g(\frac{α_{1}^{2} − w}{4} ) = w^{3} − κ_{1}w^{2} + κ_{2}w − κ_{3},

where

\begin{aligned}κ_{1} & = 3α_{1}^{2} − 8α_{2}, \\ κ_{2} & = 3α_{1}^{4} − 16α_{1}^{2}α_{2} + 16α_{1}α_{3} + 16α_{2}^{2} − 64α_{4}, \\ κ_{3} & = (α_{1}^{3} − 4α_{1}α_{2} + 8α_{3})^{2}. \end{aligned} ; (17)

Now, let Q be a quartic étale algebra over a field F of arbitrary characteristic. For x ∈ Q, let

(18)  λ_{x} = s_{2} · (x ⊗ 1 + 1 ⊗ x) ∈ Λ_{2}(Q).

Proposition 4.13. Suppose x ∈ Q is a generating element with minimal polynomial f(u) as in (13), so that the coefficient α_{1} is the trace T_{Q/F} (x) of x. Then

(a) λ_{x} + γ_{Q}(λ_{x}) = T_{Q/F} (x).

(b) γ_{Q}(λ_{x})λ_{x} ∈ R(Q) is a generating element with minimal polynomial g(v) as in (14). Moreover, if the characteristic of F is different from 2, then (λ_{x} − γ_{Q}(λ_{x}))^{2} ∈ R(Q) is a generating element with minimal polynomial h(w) as in (16).

In arbitrary characteristic, if the element κ_{3} of (17) is not zero,^{4} then λ_{x} ∈ Λ_{2}(Q) is a generating element over R(Q), with minimal polynomial

t^{2} − T_{Q/F} (x)t + γ_{Q}(λ_{x})λ_{x} ∈ R(Q)[t].

Proof. Extending scalars, we may assume that Q is split, with a basis (e_{1}, e_{2}, e_{3}, e_{4}) consisting of minimal (orthogonal) idempotents. Then R(Q) is split and e_{1} ⊗ e_{2} + e_{2} ⊗ e_{1} + e_{3} ⊗ e_{4} + e_{4} ⊗ e_{3}, e_{1} ⊗ e_{3} + e_{3} ⊗ e_{1} + e_{2} ⊗ e_{4} + e_{4} ⊗ e_{2} and e_{1} ⊗ e_{4} + e_{4} ⊗ e_{1} + e_{2} ⊗ e_{3} + e_{3} ⊗ e_{2} is a basis of R(Q) consisting of minimal idempotents. Let

x = x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}e_{4}

with x_{1}, x_{2}, x_{3}, x_{4} ∈ F. Since x generates Q, the coefficients x_{i} are pairwise distinct. Computation shows that

λ_{x} = ∑_{1≤i<j≤4}(x_{i} + x_{j})(e_{i} ⊗ e_{j} + e_{j} ⊗ e_{i}),

hence

γ_{Q}(λ_{x}) = ∑_{1≤i<j≤4}(x_{i} + x_{j})(e_{i′} ⊗ e_{j′} + e_{j′} ⊗ e_{i′})

where {i, j, i′, j′} = {1, 2, 3, 4}. It follows that

λ_{x} + γ_{Q}(λ_{x}) = (x_{1} + x_{2} + x_{3} + x_{4}) ∑_{1≤i<j≤4}(e_{i} ⊗ e_{j} + e_{j} ⊗ e_{i}) = T_{Q/F} (x).

4In characteristic 2, the condition is thus T_{Q/F} (x) ≠ 0.

Similarly

γ_{Q}(λ_{x})λ_{x} = (x_{1} + x_{2})(x_{3} + x_{4})(e_{1} ⊗ e_{2} + e_{2} ⊗ e_{1} + e_{3} ⊗ e_{4} + e_{4} ⊗ e_{3})
+ (x_{1} + x_{3})(x_{2} + x_{4})(e_{1} ⊗ e_{3} + e_{3} ⊗ e_{1} + e_{2} ⊗ e_{4} + e_{4} ⊗ e_{2})
+ (x_{1} + x_{4})(x_{2} + x_{3})(e_{1} ⊗ e_{4} + e_{4} ⊗ e_{1} + e_{2} ⊗ e_{3} + e_{3} ⊗ e_{2}).

This shows that γ_{Q}(λ_{x})λ_{x} is a root of a polynomial g whose roots in F are

y_{1} = (x_{1} + x_{2})(x_{3} + x_{4}), y_{2} = (x_{1} + x_{3})(x_{2} + x_{4}), y_{3} = (x_{1} + x_{4})(x_{2} + x_{3}).

These roots are distinct since an easy computation yields

(y_{1} − y_{2})(y_{1} − y_{3})(y_{2} − y_{3}) = − ∏_{i<j}(x_{i} − x_{j}).

Therefore, γ_{Q}(λ_{x})λ_{x} is a generator of R(Q) and g is its minimal polynomial. Similarly,

(λ_{x} − γ_{Q}(λ_{x}))^{2} = (x_{1} + x_{2} − x_{3} − x_{4})^{2}(e_{1} ⊗ e_{2} + e_{2} ⊗ e_{1} + e_{3} ⊗ e_{4} + e_{4} ⊗ e_{3})
+ (x_{1} − x_{2} + x_{3} − x_{4})^{2}(e_{1} ⊗ e_{3} + e_{3} ⊗ e_{1} + e_{2} ⊗ e_{4} + e_{4} ⊗ e_{2})
+ (x_{1} − x_{2} − x_{3} + x_{4})^{2}(e_{1} ⊗ e_{4} + e_{4} ⊗ e_{1} + e_{2} ⊗ e_{3} + e_{3} ⊗ e_{2}),

and the same arguments show that (λ_{x} − γ_{Q}(λ_{x}))^{2} is a generating element of R(Q) with minimal polynomial h as in (16) if the characteristic of F is different from 2.

Since

κ_{3} = (x_{1} + x_{2} − x_{3} − x_{4})^{2}(x_{1} − x_{2} + x_{3} − x_{4})^{2}(x_{1} − x_{2} − x_{3} + x_{4})^{2},

the condition κ_{3} ≠ 0 implies that the elements x_{i} + x_{j} for 1 ≤ i < j ≤ 4 are pairwise distinct, hence λ_{x} generates Λ_{2}(Q). Since λ_{x} + γ_{Q}(λ_{x}) = T_{E/F} (x), the minimal polynomial of λ_{x} over R(Q) is as stated in the proposition. □

Remark. Allison gives in [1, §6] another description of the algebra R(Q), for Q a quartic étale F -algebra. For x ∈ Q, he considers the image

f_{x} = φ_{2}(λ_{x}) ∈ End_{F} (⋀^{2} Q)

of λ_{x} ∈ Λ_{2}(Q) under the map φ_{2} induced by the homomorphism in (3) (see Lemma 2.3); thus

f_{x}(a ∧ b) = xa ∧ b + a ∧ xb for a, b ∈ Q.

Assuming that the characteristic of F is different from 2, Allison defines R(Q) as the span of the products f_{x} ∘f_{y}, for x, y ∈ Q of trace zero. This definition coincides with the definition in §2.1 under an isomorphism induced by φ_{2}.

4.4. Quadratic extensions of cubic étale algebras. Let A be an étale F - algebra of dimension 3, and let B be an extension of degree 2 of A. In the same spirit as the preceding subsection, we proceed to give explicit equations for generating elements of S(B/A).

Our first observation is the analogue of Proposition 1.4 through the anti-equi-valence between coverings of Γ-sets and extensions of étale F -algebras.

Proposition 4.14. There is a canonical embedding Δ(B) ↪ Ω(B/A) such that

Ω(B/A) ≃ Δ(B) ⊗_{F} S(B/A).

In the case where B (and therefore A) is split, the image of Δ(B) in Ω(B/A) is spanned by the idempotents

d = g_{0} + g_{1} + g_{2} + g_{3}  and  d′ = g′_{0} + g′_{1} + g′_{2} + g′_{3},

in the notation of Example 2.9.

In the general case, for b ∈ B we set for brevity \overline{b} = γ_{B/A}(b), and

(19)  \begin{matrix}b_{1} = s_{3}^{A} · (b ⊗ 1 ⊗ 1) = ε_{11}(b), & b′_{1} = s_{3}^{A} · (\overline{b} ⊗ 1 ⊗ 1) = ε_{21}(b) \\ b_{2} = s_{3}^{A} · (1 ⊗ b ⊗ 1) = ε_{12}(b), & b′_{2} = s_{3}^{A} · (1 ⊗ \overline{b} ⊗ 1) = ε_{22}(b) \\ b_{3} = s_{3}^{A} · (1 ⊗ 1 ⊗ b) = ε_{13}(b), & b′_{3} = s_{3}^{A} · (1 ⊗ 1 ⊗ \overline{b}) = ε_{23}(b) \end{matrix}

where ε_{ij}: B → Σ(B/A) are the embeddings of (5). Hence we have Σ(B/A) = Σ_{3}(A)[b_{1}, b′_{1}, …, b_{3}, b′_{3}], by Proposition 3.7, and 𝔖_{3} acts on Σ(B/A) through the action on Σ_{3}(A) and by permuting the b_{i} and the b′_{j}. The algebra Ω(B/A) is generated over F by all the polynomials in the b_{i} and the b′_{j} which are symmetric under 𝔖_{3}. In particular

\begin{aligned}δ_{b} & = & b_{1}b_{2}b_{3} + b′_{1}b′_{2}b_{3} + b′_{1}b_{2}b′_{3} + b_{1}b′_{2}b′_{3} \\ ω_{b} & = & b_{1} + b_{2} + b_{2} \end{aligned}

are elements of Ω(B/A).

Proposition 4.15. The element δ_{b} lies in the image of Δ(B) in Ω(B/A), and

γ_{B}(δ_{b}) = δ\overline{b}.

Moreover, the following conditions are equivalent:

(a) b generates B over A;

(b) δ_{b} generates Δ(B); Similarly, the following conditions are equivalent:

(a’) ω_{b} generates Ω(B/A) over Δ(B); (b’) (b − \overline{b})^{2} generates A.

Proof. It suffices to prove the assertions after scalar extension. We may therefore assume B is split, and use the same notation as in Example 2.9. Let

b = β_{1}f_{1} + β′_{1}f′_{1} + β_{2}f_{2} + β′_{2}f′_{2} + β_{3}f_{3} + β′_{3}f′_{3}

with β_{1}, …, β′_{3} ∈ F, hence

\overline{b} = β′_{1}f_{1} + β_{1}f′_{1} + β′_{2}f_{2} + β_{2}f′_{2} + β′_{3}f_{3} + β_{3}f′_{3}.

Computation yields

\begin{aligned}δ_{b} & = (β_{1}β_{2}β_{3} + β_{1}β′_{2}β′_{3} + β′_{1}β_{2}β′_{3} + β′_{1}β′_{2}β_{3})(g_{0} + g_{1} + g_{2} + g_{3}) \\ & + (β′_{1}β′_{2}β′_{3} + β′_{1}β_{2}β_{3} + β_{1}β′_{2}β_{3} + β_{1}β_{2}β′_{3})(g′_{0} + g′_{1} + g′_{2} + g′_{3}), \end{aligned} ; (20)

proving δ_{b} ∈ Δ_{B}. Since \overline{γ}_{B} interchanges g_{0} + ⋯ + g_{3} and g′_{0} + ⋯ + g′_{3}, it is clear that

\overline{γ}_{B}(δ_{b}) = δ\overline{b}.

We have δ_{b} ∈ F if and only if the coefficients of g_{0} +⋯+g_{3} and g′_{0} +⋯+g′_{3} in (20) above are equal, and this condition is equivalent to δ_{b} = δ\overline{b}. On the other hand, b generates B over A if and only if β_{i} ≠ β′_{i} for i = 1, 2, 3. Since

δ_{b} − δ\overline{b} = (β_{1} − β′_{1})(β_{2} − β′_{2})(β_{3} − β′_{3})(g_{0} + g_{1} + g_{2} + g_{3} − g′_{0} − g′_{1} − g′_{2} − g′_{3}), this condition holds if and only if δ_{b} ≠ δ\overline{b}. The equivalence of (a) and (b) is thus proved.

To complete the proof, let

ω_{b} = u_{0}g_{0} + u_{1}g_{1} + u_{2}g_{2} + u_{3}g_{3} + u′_{0}g′_{0} + u′_{1}g′_{1} + u′_{2}g′_{2} + u′_{3}g′_{3}.

This element generates Ω(B/A) over Δ(B) if and only if u_{0}, …, u_{3} are pairwise distinct and u′_{0}, …, u′_{3} are pairwise distinct. Computation yields

u_{0} = β_{1} + β_{2} + β_{3}, u_{1} = β_{1} + β′_{2} + β′_{3}, u_{2} = β′_{1} + β_{2} + β′_{3}, u_{3} = β′_{1} + β′_{2} + β_{3},

u′_{0} = β′_{1} + β′_{2} + β′_{3}, u′_{1} = β′_{1} + β_{2} + β_{3}, u′_{2} = β_{1} + β′_{2} + β_{3}, u′_{3} = β_{1} + β_{2} + β′_{3},

and

(u_{0} − u_{1})(u_{0} − u_{2})(u_{0} − u_{3})(u_{1} − u_{2})(u_{1} − u_{3})(u_{2} − u_{3}) =

((β_{2} − β′_{2})^{2} − (β_{3} − β′_{3})^{2})((β_{1} − β′_{1})^{2} − (β_{2} − β′_{2})^{2})((β_{1} − β′_{1})^{2} − (β_{3} − β′_{3})^{2}) =

(u′_{0} − u′_{1})(u′_{0} − u′_{2})(u′_{0} − u′_{3})(u′_{1} − u′_{2})(u′_{1} − u′_{3})(u′_{2} − u′_{3}).

Therefore, ω_{b} generates Ω(B/A) over Δ(B) if and only if (β_{1} − β′_{1})^{2}, (β_{2} − β′_{2})^{2} and (β_{3} − β′_{3})^{2} are pairwise distinct. Since

(b − \overline{b})^{2} = (β_{1} − β′_{1})^{2}e_{1} + (β_{2} − β′_{2})^{2}e_{2} + (β_{3} − β′_{3})^{2}e_{3},

this proves the equivalence of (a’) and (b’). □

Recall from [3, p. xviii] the forms T = T_{A/F}, S = S_{A/F} and N = N_{A/F} of degrees 1, 2 and 3 respectively on A, such that the generic polynomial of every element a ∈ A has the form

X^{3} − T (a)X^{2} + S(a)X − N(a) ∈ F [X].

(The form T is the trace, and N is the norm.) For a ∈ A, let a_{i} = ε_{i}(A) ∈ Σ_{3}(A). One has T (a) = a_{1} + a_{2} + a_{3}, S(a) = a_{1}a_{2} + a_{1}a_{3} + a_{2}a_{3} and N(a) = a_{1}a_{2}a_{3}.

Fix b ∈ B, and let

α_{1} = b + \overline{b} ∈ A, α_{2} = b\overline{b} ∈ A.

Computation yields

\begin{aligned}δ_{b} + δ\overline{b} & = N(α_{1}), \\ δ_{b}δ\overline{b} & = S(α_{1}^{2} − 2α_{2})T (α_{2}) − T (α_{1}^{2} − 2α_{2})T ((α_{1}^{2} − 2α_{2})α_{2}) \\ & + T ((α_{1}^{2} − 2α_{2})^{2}α_{2}) + 4N(α_{2}). \end{aligned} ; (21)

Proposition 4.16. If ω_{b} generates Ω(B/A) over Δ(B), its minimal polynomial is

X^{4} − 2T (α_{1})X^{3} + (T (α_{1}^{2}) + 2T (α_{2}) + 3S(α_{1}))X^{2}

− (4α_{b} + 2α\overline{b} + 2T (α_{1})T (α_{2}) + S(α_{1})T (α_{1}) − 3N(α_{1}))X
+ (2δ_{b} + δ\overline{b})T (α_{1}) + T (a_{1})^{2}T (α_{2}) − S(α_{1})T (α_{2})

− T (α_{1}^{2}α_{2}) + T (α_{2}^{2}) − T (α_{1})N(α_{1}) − 2S(α_{2}).

Proof. Use that in the split case, the four roots of the minimal polynomial of ω_{b} over Δ(A) are (with the notations of the proof of Proposition 4.15) the elements u_{i}, i = 0, …, 3. □

If char F ≠ 2, we may simplify the results above by a specific choice of generating element b. Let b ∈ B be such that \overline{b} = −b and assume that a = b^{2} ∈ A generates A.

Proposition 4.17. ( char F ≠ 2) With the notation above,

Δ(B) = F [δ_{b}]  and  Ω(B/A) = F [δ_{b}, ω_{b}] = Δ(B)[ω_{b}].

Moreover, the minimal polynomial of δ_{b} over F is

X^{2} − 16N(a),

and the minimal polynomial of ω_{b} over Δ(B) is

X^{4} − 2T (a)X^{2} − 2δ_{b}X + T (a^{2}) − 2S(a).

Proof. Proposition 4.15 shows that δ_{b} generates Δ(B) and that ω_{b} generates Ω(B/A) over Δ(B). This last fact can also be seen directly: the algebra Ω(B/A) is generated over F by the elementary symmetric functions in the b_{i}; since

2(b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3}) = ω_{b}^{2} − (b_{1}^{2} + b_{2}^{2} + b_{3}^{2}) = ω_{b}^{2} − T (a)

we have Ω(B/A) = F [δ_{b}, ω_{b}] = Δ(B)[ω_{b}]. The formula for the minimal polynomial of δ_{b} (resp. ω_{b}) follows from (21) (resp. Proposition 4.16). One can also repeat the proof of Proposition 4.16 with the special choice of b. □

Corollary 4.18. ( char F ≠ 2) The discriminant Δ(B) is split if and only if N(a) ∈ F^{×2}. If N(a) = ν^{2} for some ν ∈ F^{×}, then S(B/A) is generated over F by an element whose minimal polynomial is

X^{4} − 2T (a)X^{2} − 8νX + T (a^{2}) − 2S(a).

Proof. The first part readily follows from Proposition 4.17. If N(a) = ν^{2}, then Δ(B) ≃ F ×F. Let d, d′ be the minimal idempotents of Δ(B). By Proposition 4.14, we have

Ω(B/A) ≃ S(B/A) × S(B/A),

and we may identify dΩ(B/A) and d′Ω(B/A) with S(B/A). Since

dδ_{b} = ±4νd  and  d′δ_{b} = ∓4νd′,

the minimal polynomials of dω_{b} and d′λ_{b} are

X^{4} − 2T (a)X^{2} ± 8νX + T (a^{2}) − 2S(a).
□

Assume now char F = 2. Let b be a generating element for B over A with \overline{b} = b + 1, hence b^{2} + b ∈ A. Let a = b^{2} + b, i.e., using the notation ℘ for the map x ↦ x^{2} + x,

a = ℘(b) ∈ A.

Assume moreover a generates A, hence

A = F [a]  and  B = F [℘^{−1}(a)].

In contrast with Proposition 4.17, ω_{b} does not generate Ω(B/A) since (b − \overline{b})^{2} = 1 does not generate A (see Proposition 4.15). One could take for example ω_{ab} as a generator of Ω(B/A), since (ab − \overline{ab})^{2} = a^{2} (assuming that a^{2} also generates A, which for a cubic étale algebra is in general the case). However a simpler minimal polynomial is obtained for the element

μ_{b} = b_{1}b_{2} + b_{1}b_{3} + b_{2}b_{3}

(with the notations of Equation 19). Moreover we have

δ_{b} = b_{1} + b_{2} + b_{3}.

if char F = 2 and \overline{b} + b = 1.

Proposition 4.19. ( char F = 2) With the notation above,

Δ(B) = F [δ_{b}]  and  Ω(B/A) = F [δ_{b}, μ_{b}] = Δ(B)[μ_{b}].

Moreover, the minimal polynomial of δ_{b} over F is

X^{2} + X + T (a),

and the minimal polynomial of μ_{b} over Δ(B) is

X^{4} + X^{3} + (δ_{b}^{2} + 1)X^{2} + (δ_{b}^{2} + S(a) + 1)X + (δ_{b} + S(a) + 1)S(a) + N(a).

Proof. Since b generates B over A, Proposition 4.15 shows that Δ(B) = F [δ_{b}], and Equations (21) yield the minimal polynomial of δ_{b}.

To prove the rest, we extend scalars and assume B is split. Using the same notation as in Example 2.9, we have

b = β_{1}f_{1} + (β_{1} + 1)f′_{1} + β_{2}f_{2} + (β_{2} + 1)f′_{2} + β_{3}f_{3} + (β_{3} + 1)f′_{3}

for some β_{1}, β_{2}, β_{3} ∈ F. Then, letting

d = g_{0} + g_{1} + g_{2} + g_{3}  and  d′ = g′_{0} + g′_{1} + g′_{2} + g′_{3}

be the minimal idempotents of Δ(B) ⊂ Ω(B/A), we have

δ_{b} = (β_{1} + β_{2} + β_{3})d + (β_{1} + β_{2} + β_{3} + 1)d′,

and

μ_{b} = v_{0}g_{0} + v′_{0}g′_{0} + v_{1}g_{1} + v′_{1}g′_{1} + v_{2}g_{2} + v′_{2}g′_{2} + v_{3}g_{3} + v′_{3}g′_{3}

where

v_{0} = β_{1}β_{2} + β_{1}β_{3} + β_{2}β_{3},  v′_{0} = v_{0} + 1,
v_{1} = β_{1}β_{2} + β_{1}β_{3} + β_{2}β_{3} + β_{2} + β_{3} + 1,  v′_{1} = v_{1} + 1,
v_{2} = β_{1}β_{2} + β_{1}β_{3} + β_{2}β_{3} + β_{1} + β_{3} + 1,  v′_{2} = v_{2} + 1,
v_{3} = β_{1}β_{2} + β_{1}β_{3} + β_{2}β_{3} + β_{1} + β_{2} + 1,  v′_{3} = v_{3} + 1.

Then

∏_{0≤i<j≤3}(v_{i} − v_{i}) = ∏_{1≤i<j≤3}(β_{i} − β_{j})^{2}.

Since b generates B over A, the elements β_{1}, β_{2}, β_{3} are pairwise distinct, hence this equality shows that v_{0}, …, v_{3} are pairwise distinct. Similarly, v′_{0}, …, v′_{3} are pairwise distinct, hence μ_{b} generates Ω(B/A) over Δ(B). We have

T (a) = (β_{1}^{2} + β_{1}) + (β_{2}^{2} + β_{2}) + (β_{3}^{2} + β_{3}),
S(a) = (β_{1}^{2} + β_{1})(β_{2}^{2} + β_{2}) + (β_{1}^{2} + β_{1})(β_{3}^{2} + β_{3}) + (β_{2}^{2} + β_{2})(β_{3}^{2} + β_{3}),
N(a) = (β_{1}^{2} + β_{1})(β_{2}^{2} + β_{2})(β_{3}^{2} + β_{3}),

and brute force computation shows that v_{0}, v_{1}, v_{2} and v_{3} are roots of

X^{4} + X^{3} + (β_{1}^{2} + β_{2}^{2} + β_{3}^{2} + 1)X^{2} + (β_{1}^{2} + β_{2}^{2} + β_{3}^{2} + S(a) + 1)X

+ (β_{1} + β_{2} + β_{3} + S(a) + 1)S(a) + N(a).

Similarly, v′_{0}, v′_{1}, v′_{2} and v′_{3} are roots of

X^{4} + X^{3} + (β_{1}^{2} + β_{2}^{2} + β_{3}^{2})X^{2} + (β_{1}^{2} + β_{2}^{2} + β_{3}^{2} + S(a))X

+ (β_{1} + β_{2} + β_{3} + S(a))S(a) + N(a), hence the proof is complete. □

Corollary 4.20. ( char F = 2) With the same notation as in Proposition 4.19, the discriminant Δ(B) is split if and only if T (a) ∈ ℘(F). If T (a) = ℘(ν) for some ν ∈ F, then S(B/A) is generated over F by an element whose minimal polynomial is

X^{4} + X^{3} + ν^{2}X^{2} + (ν^{2} + S(a))X + (ν + S(a))S(a) + N(a).

Proof. The first part readily follows from Proposition 4.19. If T (a) = ℘(ν), then Δ(B) = F × F. Let d, d′ be the minimal idempotents of Δ(B) ⊂ Ω(B/A). We may assume d = δ_{b} + ν and d′ = δ_{b} + ν + 1, hence

dδ_{b} = (ν + 1)d.

As in the proof of Corollary 4.18, we may identify dΩ(B/A) and d′Ω(B/A) with S(B/A), and it follows from Proposition 4.19 that the minimal polynomial of dμ_{b} is as stated. □

Combining the results of subsections 4.3 and 4.4 we get:

Proposition 4.21. Let Q be a quartic étale algebra.

(a) If char F ≠ 2, let x ∈ Q be a generator such that T_{Q/F} (x) = 0. There exists an isomorphism ϕ_{1} : Q ⟶^{∼} S(Λ_{2}(Q)/ R(Q)) such that ϕ_{1}(x) = ω_{λ_{x}}.

(b) If char F = 2, let x ∈ Q be a generator such that T_{Q/F} (x) = 1. There exists an isomorphism ϕ_{2} : Q ⟶^{∼} S(Λ_{2}(Q)/ R(Q)) such that ϕ_{2}(x) = μ_{λ_{x}}.

5. Special actions on four elements

As in the preceding sections, Γ denotes a profinite group. The constructions on Γ-sets given in §1.1 take a special form when the Γ-action has some particular properties. For instance, if the action on a set X is not transitive, then the orbits X_{1}, …, X_{r} under Γ yield a Γ-set decomposition

X = X_{1} ∐ … ∐ X_{r}.

Even if the Γ-action on X is transitive, the induced action on Σ_{n}(X) may not be transitive.

Proposition 5.1. Let X be a Γ-set with |X| = n, and let α: Γ → 𝔖_{X} be the action of Γ. If (𝔖_{X} : α(Γ)) = r, there is a Γ-set decomposition

(22)  Σ_{n}(X) = Ω_{1} ∐ … ∐ Ω_{r}.

Each Γ-set Ω_{i} is a G_{i}-torsor for some subgroup G_{i} ⊂ 𝔖_{n} isomorphic to α(Γ), and the subgroups G_{i} are conjugate in 𝔖_{n}. Moreover, the Γ-sets Ω_{1}, …, Ω_{r} are isomorphic.

Proof. The Γ-orbits of Σ_{n}(X) yield the decomposition (22). To see that each Ω_{i} is a G_{i}-torsor, recall that for (x_{1}, …, x_{n}) ∈ Σ_{n}(X), γ ∈ Γ and σ ∈ 𝔖_{n}, we have by definition

γ(x_{1}, …, x_{n}) = (α(γ)(x_{1}), …, α(γ)(x_{n})) and (x_{1}, …, x_{n})^{σ} = (x_{σ(1)}, …, x_{σ(n)}).

For each γ ∈ Γ, there is a unique σ ∈ 𝔖_{n} such that

γ(x_{1}, …, x_{n}) = (x_{1}, …, x_{n})^{σ},

and the map γ ↦ σ defines a homomorphism Γ → 𝔖_{n} which depends on the choice of (x_{1}, …, x_{n}) and factors through α to yield an injection α(Γ) ↪ 𝔖_{n}. If (x_{1}, …, x_{n}) ∈ Ω_{i}, let G_{i} ⊂ 𝔖_{n} be the image of this map. Then Ω_{i} is a G_{i}-torsor. Moreover, if (x_{1}, …, x_{n}) ∈ Ω_{i} and (y_{1}, …, y_{n}) ∈ Ω_{j}, there exists σ ∈ 𝔖_{n} such that

(y_{1}, …, y_{n}) = (x_{1}, …, x_{n})^{σ}.

Conjugation by σ maps G_{j} to G_{i} and the action of σ defines an isomorphism of

Γ-sets Ω_{i} ⟶^{∼} Ω_{j}. □

Remark. The 𝔖_{n}-torsor Σ_{n}(X) can be obtained as the induced torsor Ind_{G_{1}}^{𝔖_{n}} Ω_{1}, by mimicking the construction in [3, (18.17)].

Note that if the Γ-action on X is transitive, then the map X ⟵^{π_{n}} Σ_{n}(X) restricts to each Ω_{i} to define a covering X ← Ω_{i}. Extending Definition 3.2, this covering may be regarded as a G_{i}-Galois closure of X.

Taking for Γ the absolute Galois group of a field F, and using the anti-equivalence Ét_{F} ≡ Set_{Γ} of Section 2, we may adapt the construction above to étale algebras. Disjoint unions of Γ-sets correspond to direct product decompositions of algebras, hence an étale F -algebra is a field if and only if the Γ-action on X(E) is transitive. If dim E = n, Proposition 5.1 thus yields a direct product decomposition of the 𝔖_{n}-Galois closure Σ_{n}(E) into isomorphic fields

Σ_{n}(E) ≃ L_{1} × ⋯ × L_{r}.

Each L_{i} is a Galois extension of F with Galois group G_{i} ⊂ 𝔖_{n} isomorphic to the image of the action Γ → 𝔖_{X(E)}. If E is a field, each L_{i} can be regarded as a Galois closure of E/F, see [3, (18.22)].

In the rest of this section, we consider the particular case where n = 4. To determine the various possibilities for the image of the action Γ → 𝔖_{4}, we list the subgroups of 𝔖_{4}.

Proposition 5.2. In the symmetric group 𝔖_{4}, • the alternating group 𝔄_{4} is the unique subgroup of order 12;

• there are four subgroups of order 6; they are conjugate to 𝔖_{3};

• there are three subgroups of order 8; they are pairwise conjugate and iso-

morphic to the dihedral group D_{4}. Moreover, every proper subgroup of 𝔖_{4} is contained in at least one of the subgroups listed above.

Proof. Any subgroup of index 2 in 𝔖_{4} must contain all the Sylow 3-subgroups of 𝔖_{4}. Since these Sylow subgroups are generated by the cycles of length 3, the first claim is clear.

A subgroup of order 6 in 𝔖_{4} cannot be transitive on {1, 2, 3, 4}. On the other hand, it has an orbit of 3 elements since it contains a Sylow 3-subgroup, hence it must be the isotropy group of one of 1, 2, 3, or 4.

The dihedral group D_{4} acts on the four vertices of a square, hence it may be considered as a subgroup of 𝔖_{4}. It is then identified with a 2-Sylow subgroup of 𝔖_{4}, and all the 2-Sylow subgroups are conjugate.

Finally, let G ⊂ 𝔖_{4} be a subgroup. If its order is divisible by 3, then it is 3, 6 or 12, hence G is contained in 𝔄_{4} or in a conjugate of 𝔖_{3}. If its order is a power of 2, then G is contained in a 2-Sylow subgroup. □

In the following subsections, we examine the additional information on a Γ-set X with |X| = 4 (or on a quartic étale F -algebra Q) when the Γ-action factors through a subgroup 𝔖_{3} or D_{4}. We then collect the information to obtain a classification of quartic étale F -algebras in §5.3.

5.1. Action through 𝔖_{3}. Suppose first the action of Γ leaves an element x ∈ X invariant. It then preserves a disjoint union decomposition

X = {x} ∐ R,

where |R| = 3. The 2-element subsets of X containing x are in one-to-one correspondence with R, hence Λ_{2}(X) decomposes as

Λ_{2}(X) = R ∐ Λ_{2}(R).

Moreover, the “complementary subset” involution γ_{X} on Λ_{2}(X) interchanges R and Λ_{2}(R) and defines an isomorphism R ≃ Λ_{2}(R). Therefore, we have canonical isomorphisms

Λ_{2}(X) ≃ R ∐ R and R (X) ≃ R.

(See also §4.2.)

Assuming Γ is the absolute Galois group of a field F, we may translate the results above in the framework of étale F -algebras.

Proposition 5.3. Let Q be a quartic étale F -algebra. If the Γ-action on X(Q) factors through a subgroup 𝔖_{3} ⊂ 𝔖_{X(Q)}, then there is a cubic field extension L/F such that

Q ≃ F × L, Λ_{2}(Q) ≃ L × L, and R(Q) ≃ L.

Moreover, the following conditions are equivalent:

(a) the Γ-action factors through a cyclic subgroup C_{3};

(b) the extension L/F is Galois (hence cyclic);

(c) Δ(Q) ≃ F × F.

5.2. Action through D_{4}. Suppose now that the action of Γ factors through a Sylow 2-subgroup of 𝔖_{X}, i.e. through a dihedral subgroup D_{4}. Since the Sylow 2subgroups of 𝔖_{X} are the isotropy groups of partitions of X into 2-element subsets, there is such a partition which is invariant under Γ. This observation characterizes the case where Γ acts through D_{4}:

Proposition 5.4. For a set X ∈ Set_{Γ}^{4}, the following conditions are equivalent: (a) the Γ-action factors through a Sylow 2-subgroup of 𝔖_{X};

(b) the Γ-action leaves a point of R(X) fixed;

(c) R(X) ≃ {∗} ∐ Δ(X);

(d) X is a double covering of a set of two elements, i.e. there exists a map (D ← X) ∈ Cov_{Γ}^{2≀2}.

Proof. The points of R(X) are the partitions of X into 2-element subsets, hence (a) ⟺ (b). The implication (c)⇒(b) is clear, and (b)⇒(c) follows from Proposition 4.1. If D = {{x_{1}, x_{2}}, {x_{3}, x_{4}}} ∈ R(X) is fixed under Γ, then the canonical map D ← X which carries x_{1}, x_{2} to {x_{1}, x_{2}} and x_{3}, x_{4} to {x_{3}, x_{4}} is a double covering, hence (b)⇒(d). Finally, (d)⇒(a) follows from 𝔖_{2} ≀ 𝔖_{2} ≃ D_{4}. □

The following proposition establishes the existence of a “dual” Γ-set X̌:

Proposition 5.5. If the equivalent conditions of Proposition 5.4 hold, then there exists a Γ-set X̌ ∈ Set_{Γ}^{4}, with Γ-action through a Sylow 2-subgroup of 𝔖X̌, such that

Λ_{2}(X) ≃ Δ(X̌) ∐ X̌,  Λ_{2}(X̌) ≃ Δ(X) ∐ X,
R(X) ≃ {Δ(X̌)} ∐ Δ(X),  R(X̌) ≃ {Δ(X)} ∐ Δ(X̌).

Moreover, X is a double covering of Δ(X̌), and X̌ is a double covering of Δ(X).

If the Γ-action on Δ(X) is not trivial, the Γ-set X̌ is canonically determined. If the Γ-actions on Δ(X) and Δ(X̌) are not trivial, then there is a canonical isomorphism

ˇX̌ = X.

Proof. Let D ∈ R(X) be a fixed point of Γ. Define X̌ as the complementary subset in Λ_{2}(X) of the fiber ε^{−1}(D) under the canonical map R(X) ⟵^{ε} Λ_{2}(X). The set X̌ is thus canonically determined if Γ has a unique fixed point D ∈ R(X), or, equivalently by Proposition 5.4, if the Γ-action on Δ(X) is not trivial.

We proceed to prove that X̌ satisfies the stated properties. To clarify the discussion, we use geometric language. If D = {{x_{1}, x_{2}}, {x_{3}, x_{4}}}, we identify X with the set of vertices of a square, letting {x_{1}, x_{2}} and {x_{3}, x_{4}} be the pairs of opposite vertices. We may thus identify D to the set of diagonals of the square, and we have a decomposition

(23)  Λ_{2}(X) = D ∐ X̌

where X̌ is the set of pairs of adjacent vertices, which may be identified with the set of edges of the square. (Note that X̌ may also be viewed as the dual square of X in the sense of polytope theory.) There is a “dual” decomposition

(24)  Λ_{2}(X̌) = M ∐ X,

where X is identified with the set of pairs of adjacent edges (by mapping every such pair to their common vertex) and M is the set of pairs of parallel edges, which may be identified with the medians of the square. The “complementary subset” involutions γ_{X} and γX̌ preserve the decompositions (23) and (24), and the set of orbits of X̌ (resp. X) under γ_{X} (resp. γX̌) can be identified with M (resp. D), hence

(25)  R(X) = {D} ∐ M  and  R(X̌) = {M} ∐ D,

and there are natural maps D ← X and M ← X̌ which show X and X̌ are double coverings of D and M respectively. By Proposition 4.1, Equations (25) yield canonical isomorphisms

M = Δ(X)  and  D = Δ(X̌).

If the Γ-action on Δ(X̌) is not trivial, then M is the unique fixed point of R(X̌), and X ⊂ Λ_{2}(X̌) is the complementary subset of the fiber of M under the canonical map R(X̌) ← Λ_{2}(X̌), hence ˇ X̌ = X. This completes the proof. □

We may use the Γ-set X̌ to obtain information on the Γ-action on X, as follows:

Proposition 5.6. Let X ∈ Set_{Γ}^{4} be a Γ-set satisfying the equivalent properties of Proposition 5.4, and suppose the Γ-action on Δ(X) is not trivial, so the dual set X̌ is uniquely determined. The following properties are equivalent:

(a) the Γ-action on X factors through a cyclic subgroup C_{4};

(b) X̌ ≃ X;

(c) Δ(X̌) ≃ Δ(X).

Proof. If the action of Γ factors through C_{4}, we may regard X as the set of vertices of an oriented square, and use the orientation to define a canonical isomorphism X ⟶^{∼} X̌, proving (a)⇒(b). Since the implication (b)⇒(c) is clear, it only remains to prove (c)⇒(a). If the image of Γ under the action contains the Vierergruppe 𝔙_{X}, then there is an element in Γ which acts trivially on M and non-trivially on D, hence Δ(X) ≄ Δ(X̌). Similarly, if some element of Γ acts by a single transposition on X, then it acts trivially on D and non-trivially on M, hence Δ(X) ≄ Δ(X̌). Therefore, (c) implies that the image of the action of Γ contains at most cycles of length 4 and one element of 𝔙_{X}. □

Remark. The Γ-set D ∗ M = Δ(X) ∗ Δ(X̌) can be identified with the set of orientations of the square.

For the following proposition, recall that the dihedral group D_{4} contains two non-conjugate elementary abelian subgroups C_{2} × C_{2}. Viewing D_{4} as a subgroup of 𝔖_{4}, one of these subgroups is 𝔙 (= D_{4} ∩ 𝔄_{4}). The other one is generated by two disjoint transpositions; it is not transitive on {1, 2, 3, 4}.

Proposition 5.7. Let X ∈ Set_{Γ}^{4} be a Γ-set satisfying the equivalent properties of Proposition 5.4, and suppose the Γ-action on Δ(X) is not trivial, so the dual set X̌ is uniquely determined. The following properties are equivalent:

(a) the Γ-action on X factors through an elementary abelian subgroup C_{2}×C_{2} ≠ 𝔙_{X};

(b) the Γ-action on X̌ factors through 𝔙X̌;

(c) the Γ-action on Δ(X̌) is trivial.

The proof is left to the reader.

Finally, we consider the case where the Γ-action on X factors through 𝔙_{X}.

Proposition 5.8. For a Γ-set X ∈ Set_{Γ}^{4}, the following conditions are equivalent: (a) the Γ-action on X factors through the Vierergruppe 𝔙_{X};

(b) the Γ-action on R(X) is trivial;

(c) the Γ-set Λ_{2}(X) has a decomposition into 2-element subsets stable under the canonical involution of Λ_{2}(X)/ R(X),

Λ_{2}(X) = D_{1} ∐ D_{2} ∐ D_{3};

(d) X satisfies the equivalent conditions of Proposition 5.4 and Γ acts trivially

on Δ(X). Moreover, if these conditions hold, then the Γ-action on D_{1} ∗ D_{2} ∗ D_{3} is trivial.

Proof. The Vierergruppe can be defined as the subgroup of 𝔖_{X} which leaves invariant all the partitions of X into 2-element subsets, hence (a) ⟺ (b). The equivalence of (b) and (c) is clear: take for D_{1}, D_{2} and D_{3} the fibers of the canonical map R(X) ← Λ_{2}(X). The equivalence (b) ⟺ (d) readily follows from Proposition 5.4.

If the equivalent conditions of the proposition hold, then the set X̌ of Proposition 5.5 can be arbitrarily chosen as D_{1} ∐ D_{2}, D_{1} ∐ D_{3} or D_{2} ∐ D_{3}. If we choose X̌ = D_{1} ∐ D_{2}, Proposition 5.5 yields Δ(X̌) = D_{3}. On the other hand, it is easily checked that

Δ(D_{1} ∐ D_{2}) ≃ D_{1} ∗ D_{2},

hence D_{1} ∗ D_{2} ≃ D_{3} and therefore the Γ-action on D_{1} ∗ D_{2} ∗ D_{3} is trivial. □

Taking for Γ the absolute Galois group of a field F, we may translate in terms of étale F -algebras the results of this subsection, by using the anti-equivalence Ét_{F} ≡ Set_{Γ} of §2. By a quartic 2-algebra we mean an étale algebra which is a quadratic extension of a quadratic étale algebra. These algebras can be characterized through Proposition 5.4:

Proposition 5.9. For a quartic F -algebra Q, the following conditions are equivalent:

(a) the Γ-action on X(Q) factors through a Sylow 2-subgroup of 𝔖_{X(Q)};

(b) R(Q) is not a field;

(c) R(Q) ≃ F × Δ(Q);

(d) Q is a quartic 2-algebra.

Proposition 5.5 proves for every quartic 2-algebra Q the existence of a “dual” quartic 2-algebra Q̌, which is canonically determined if Δ(Q) is not split. This algebra is a quadratic extension of Δ(Q), and Q is a quadratic extension of Δ(Q̌). Moreover, Q and Q̌ satisfy the following relations:

Λ_{2}(Q) ≃ Δ(Q̌) × Q̌,  Λ_{2}(Q̌) ≃ Δ(Q) × Q,
R(Q) ≃ F × Δ(Q),  R(Q̌) ≃ F × Δ(Q̌).

We record a few special cases:

Proposition 5.10. Let Q be a quartic 2-algebra over F.

(1) If Q is a cyclic field extension of F, then Q̌ ≃ Q, hence Q is a quadratic extension of Δ(Q), and

Λ_{2}(Q) ≃ Δ(Q) × Q, R(Q) ≃ F × Δ(Q).

(2) If Q = K_{1}×K_{2}, where K_{1} and K_{2} are non-isomorphic quadratic F -algebras, then Q̌ ≃ K_{1} ⊗_{F} K_{2}, Δ(Q) ≃ K_{1} ∗ K_{2}, and

Λ_{2}(Q) ≃ F × F × (K_{1} ⊗_{F} K_{2}), R(Q) ≃ F × (K_{1} ∗ K_{2}).

(3) If Q = K_{1} ⊗_{F} K_{2}, where K_{1} and K_{2} are quadratic field extensions of F, then one may take Q̌ = K_{1} × K_{2}, or K_{1} × (K_{1} ∗ K_{2}), or K_{2} × (K_{1} ∗ K_{2}). Moreover, Δ(Q) is split, and

Λ_{2}(Q) ≃ K_{1} × K_{2} × (K_{1} ∗ K_{2}), R(Q) ≃ F × F × F.

These results are easily derived from Propositions 5.6, 5.7 and 5.8. Note that split quadratic algebras are allowed in (2), and that the case where Q = K × K for some quadratic field extension K of F is covered by (3) since K × K ≃ K ⊗_{F} K.

Since Λ_{2}(Q) ≃ Δ(Q̌)×Q̌, we may use the computations of §4.3 to give an explicit description of Q̌.

Proposition 5.11. Let Q be a quartic 2-algebra over F, and let K ⊂ Q be a quadratic étale F -algebra. Denote by the canonical involution of K over F.

(1) Suppose char F ≠ 2 and Q = K(\sqrt{y}) where y ∈ K generates K. Then

Δ(Q) ≃ F (\sqrt{N_{K/F} (y)}) and Q̌ ≃ F (\sqrt{y} + \sqrt{\overline{y}}) (and Δ(Q̌) ≃ K).

(2) Suppose char F = 2 and Q = K(℘^{−1}(y)) where y ∈ K generates K. Then

Δ(Q) ≃ F (℘^{−1}(T_{K/F} (y))) and Q̌ ≃ F (℘^{−1}(y)℘^{−1}(\overline{y})) (and Δ(Q̌) ≃ K).

In the proof below, we write simply T (y), N(y) for T_{K/F} (y) and N_{K/F} (y).

Proof. (1) By hypothesis, the element x = \sqrt{y} generates Q over F. Its minimal polynomial is

(26)  u^{4} − T (y)u^{2} + N(y) ∈ F [u].

Let λ_{x} ∈ Λ_{2}(Q) be defined as in (18). By Proposition 4.13, γ_{Q}(λ_{x})λ_{x} generates R(Q) and its minimal polynomial is

t((t + T (y))^{2} − 4N(y)).

Therefore,

(27)  R(Q) ≃ F × F (\sqrt{N(y)}),

determining Δ(Q). Moreover, Proposition 4.13 also shows that γ_{Q}(λ_{x}) = −λ_{x}, hence regarding (27) as an identification,

λ_{x}^{2} = (0, T (y) − 2\sqrt{N(y)}).

Therefore, the projection λ̌_{x} of λ_{x} to Q̌ under the isomorphism Λ_{2}(Q) ≃ Δ(Q̌) × Q̌ satisfies

(28)  λ̌_{x}^{2} = T (y) − 2\sqrt{N(y)} = y + \overline{y} − 2\sqrt{y\overline{y}}.

If the minimal polynomial (26) of x has no root in F, computation shows that T (y) − 2\sqrt{N(y)} is not a square in Δ(Q), hence

Q̌ = F (λ̌_{x}).

The proof is complete, since (28) shows that we may identify λ̌_{x} with \sqrt{y} + \sqrt{\overline{y}}, determining the square roots in such a way that \sqrt{y}\sqrt{\overline{y}} = −\sqrt{N(y)}.

If the minimal polynomial (26) has a root in F, then Q has a factor F and we are in the situation of Proposition 5.10(2) with K_{1} or K_{2} split. This case is left to the reader.

(2) Suppose now char F = 2. The element x = y℘^{−1}(y) generates Q with minimal polynomial

u^{4} + T (y)u^{3} + (T (y)^{3} + T (y)N(y) + N(y))u^{2} + T (y)^{2}N(y)u + N(y)^{3} ∈ F [u].

Consider again the element λ_{x} ∈ Λ_{2}(Q) defined in (18). By Proposition 4.13, γ_{Q}(λ_{x})λ_{x} generates R(Q) and has minimal polynomial

(t − N(y))(t^{2} − N(y)t + T (y)^{6} + T (y)^{2}N(y) + T (y)N(y)^{2}).

If w is a root of the quadratic factor, then

℘(N(y)^{−1}(w − T (y)^{3} − T (y)N(y))) = T (y),

hence

R(Q) ≃ F × F (℘^{−1}(T (y))).

Proposition 4.13 also shows that the projection λ̌_{x} of λ_{x} onto Q̌ satisfies

(29)  λ̌_{x}^{2} + T (y)λ̌_{x} + w = 0.

If ℘^{−1}(y) and ℘^{−1}(\overline{y}) are determined in such a way that ℘^{−1}(y) + ℘^{−1}(\overline{y}) = ℘^{−1}(T (y)), computation shows that ℘^{−1}(T (y))(T (y) + ℘^{−1}(y)℘^{−1}(\overline{y})) also satisfies (29), hence we may identify Q̌ with F (℘^{−1}(y)℘^{−1}(\overline{y})). □

We refer to [5] for a description of quartic 2-extensions of fields in characteristic different from 2.

5.3. Classification of quartic algebras. Combining the results of §§5.1 and 5.2, we obtain a classification of quartic étale F -algebras Q based on the action of the absolute Galois group Γ of F on X(Q). We summarize the various possibilities for Q, Δ(Q), R(Q) and Λ_{2}(Q) in the table below. In this table, α(Γ) ⊂ 𝔖_{X(Q)} ≃ 𝔖_{4} is the image of the Γ-action. The letters N and L are used for sextic and cubic separable field extensions of F, and K, Ǩ, K_{1}, K_{2} for quadratic separable field extensions of F. A quartic separable field extension is called an 𝔖_{4}-quartic (resp. 𝔄_{4}-quartic) if its Galois closure has Galois group isomorphic to 𝔖_{4} (resp. 𝔄_{4}).

α(Γ)  Q  Δ(Q)  R(Q)  Λ_{2}(Q)
{1}  F^{4}  F^{2}  F^{3}  F^{6}
C_{2} ⊄ 𝔙  F^{2} × K  K  F × K  F^{2} × K^{2}
C_{2} ⊂ 𝔙  K × K  F^{2}  F^{3}  F^{2} × K^{2}
C_{3}  F × L, L cyclic F^{2}  L  L × L
C_{2} × C_{2} ⊄ 𝔙 K_{1} × K_{2}  K_{1} ∗ K_{2} F × K_{1} ∗ K_{2} F^{2} × (K_{1} ⊗ K_{2})
𝔙  K_{1} ⊗ K_{2}  F^{2}  F^{3}  K_{1} × K_{1} × K_{1} ∗ K_{2}
C_{4}  cyclic  K ⊂ Q F × K  K × Q
S_{3}  F × L  K  L  L × L
D_{4}  Q ⊃ Ǩ  K  F × K  Ǩ × Q̌, Q̌ ⊃ K
𝔄_{4}  𝔄_{4}-quartic  F^{2}  L cyclic  N ⊃ L
𝔖_{4}  𝔖_{4}-quartic  K  L S_{3}-cubic  N ⊃ L

6. Cyclic quartic algebras

Let F be an arbitrary field with absolute Galois group Γ = Gal(F_{s}/F). Quartic étale F -algebras Q such that the Γ-action on X(Q) factors through a cyclic group C_{4} can be endowed with the structure of a C_{4}-Galois algebra. (In the table of §5.3, they can be found in the lines α(Γ) = {1}, α(Γ) = C_{2} ⊂ 𝔙 and α(Γ) = C_{4}.) Fixing a generator of C_{4} (or, equivalently, choosing an isomorphism C_{4} ≃ ℤ/4ℤ), we may consider a C_{4}-Galois F -algebra as a pair (Q, ν) where Q is a quartic étale F -algebra and ν is an F -algebra automorphism of Q such that

{x ∈ Q | ν(x) = x} = F.

The automorphism ν then satisfies ν^{4} = Id, and it yields the action on Q of the generator of C_{4}. An isomorphism of C_{4}-Galois F -algebras β: (Q, ν) → (Q′, ν′) is an isomorphism β: Q ⟶^{∼} Q′ such that ν′ ∘ β = β ∘ ν. Let Cycl_{4}(F) be the set of isomorphism classes of C_{4}-Galois F -algebras. As observed in §3.5, there is a canonical bijection

Cycl_{4}(F) ≃ H^{1}(Γ, C_{4}).

If C_{4} is embedded in 𝔖_{4}, the corresponding map in cohomology H^{1}(Γ, C_{4}) → H^{1}(Γ, 𝔖_{4}) maps the isomorphism class of (Q, ν) to the isomorphism class of Q.

Since C_{4} is an abelian group, the set H^{1}(Γ, C_{4}) is an abelian group. The group structure on Cycl_{4}(F) is induced by the following composition law (see [2]):

(Q, ν) ⋆ (Q′, ν′) = ((Q ⊗ Q′)^{ν^{−1}⊗ν′}, ν ⊗ Id).

The class of the split algebra F^{4} with the cyclic permutation of factors is the neutral element. The squaring map ρ: C_{4} → 𝔖_{2} fits into an exact sequence

1 → 𝔖_{2} ⟶^{ι} C_{4} ⟶^{ρ} 𝔖_{2} → 1.

Since H^{1}(Γ, 𝔖_{2}) ≃ Quad(F) (see §3.4), the induced exact sequence in cohomology takes the form

(30)  1 → Quad(F) ⟶^{ι^{1}} Cycl_{4}(F) ⟶^{ρ^{1}} Quad(F).

The map ι^{1} is induced by K ↦ (K × K, ν) where ν(x, y) = (y, γ_{K}(x)), and the map ρ^{1} carries every C_{4}-Galois algebra (Q, ν) to its discriminant Δ(Q) (which is isomorphic to the quadratic subalgebra Q^{ν^{2}}, see Proposition 5.10).

Remark. The algebra K × K contains K and F × F as quadratic subalgebras. However, Galois theory shows that if (Q, ν) is a C_{4}-Galois algebra and Q is a field, then Q contains a unique quadratic extension of F.

In the rest of this section, we give an explicit description of H^{1}(Γ, C_{4}) and use it to parametrize C_{4}-Galois algebras up to isomorphism. The description depends in an essential way on whether the characteristic is 2 or not.

6.1. Characteristic not 2. If char F ≠ 2, the group μ_{4}(F_{s}) ⊂ F_{s} of fourth roots of unity is cyclic of order 4. Let S = F [X]/(X^{2} + 1). Twisting the Γ-action on μ_{4}(F_{s}) by a cocycle whose image in H^{1}(Γ, 𝔖_{2}) defines S, we obtain a Γ-module μ_{4[S]} with trivial Γ-action. Thus, μ_{4[S]} ≃ C_{4}, and C_{4}-Galois F -algebras are also classified by H^{1}(Γ, μ_{4[S]}). To give an explicit description of this group, consider the homomorphism

Υ: F^{×} × S^{×} → F^{×} × S^{×}

defined by

Υ(ℓ, z) = (N_{S/F} (z), ℓz^{2}).

Proposition 6.1. There is a canonical isomorphism

H^{1}(Γ, μ_{4[S]}) ≃ (F^{×} × S^{×})/Υ(F^{×} × S^{×}).

Proof. Let i = \sqrt{−1} ∈ S be the image of X. We may identify μ_{4[S]} with

{(1, 1), (1, −1), (−1, 1 ⊗ i), (−1, −1 ⊗ i)} ⊂ F_{s}^{×} × (F_{s} ⊗ S)^{×},

which is the kernel of the map Υ extended to F_{s}. The proposition follows from the cohomology exact sequence associated with

1 → μ_{4[S]} → F_{s}^{×} × (F_{s} ⊗ S)^{×} ⟶^{Υ} F_{s}^{×} × (F_{s} ⊗ S)^{×} → 1,

since Hilbert’s Theorem 90 and Shapiro’s lemma yield H^{1}(Γ, F_{s}^{×} × (F_{s} ⊗ S)^{×}) = 1. □

Remark. Another description of H^{1}(Γ, μ_{4[S]}) is given in [3, (30.13)].

It follows from Proposition 6.1 that C_{4}-Galois algebras are classified by the group

(F^{×} × S^{×})/Υ(F^{×} × S^{×}).

(See [2] for a proof without cohomology and, more generally for a class of commutative rings in which 2 is invertible.) We give an explicit description of this correspondence.

Let i = \sqrt{−1} ∈ S. For λ ∈ F^{×} and s = s_{1} + is_{2} ∈ S^{×}, let d = N_{S/F} (s) = s_{1}^{2} + s_{2}^{2} and let

Q_{λ,s} = F [W, X, Y]/I_{λ,s}

where I_{λ,s} is the ideal generated by

W^{2} − d, X^{2} − \frac{λ}{2}(d + s_{1}W), Y^{2} − \frac{λ}{2}(d − s_{1}W), XY − \frac{λ}{2}s_{2}W.

The automorphism ν of F [W, X, Y] defined by

ν(W) = −W, ν(X) = Y, ν(Y) = −X

preserves I_{λ,s} and induces an automorphism of Q_{λ,s} which we denote by ν_{λ,s}.

Proposition 6.2. For any λ ∈ F^{×} and s ∈ S^{×}, the pair (Q_{λ,s}, ν_{λ,s}) is a C_{4}-Galois F -algebra. The map (λ, s) ↦ (Q_{λ,s}, ν_{λ,s}) induces a group isomorphism

Φ: F^{×} × S^{×}/{(N_{S/F} (z), ℓz^{2}) | ℓ ∈ F^{×}, z ∈ S^{×}} ⟶^{∼} Cycl_{4}(F).

Proof. We first show that Q_{λ,s} is a quartic étale F -algebra. Let ω, ξ, η ∈ Q_{λ,s} be the images of W, X, Y respectively. The algebra F [ω] is quadratic, F [ω] ≃ F [\sqrt{d}].

If s_{2} ≠ 0, then d ≠ s_{1}^{2}, hence d + s_{1}ω is invertible. Computation shows (ξ^{−1} \frac{λ}{2}s_{2}ω)^{2} = \frac{λ}{2}(d − s_{1}ω), so

Q_{λ,s} = F [ω, ξ] ≃ F [\sqrt{d}][\sqrt{\frac{λ}{2}(d + s_{1}\sqrt{d})}].

If s_{2} = 0, then d = s_{1}^{2}, hence F [ω] ≃ F × F and we may identify ξ and η to s_{1}(\sqrt{λ}, 0) and s_{1}(0, \sqrt{λ}) in

Q_{λ,s} ≃ F [\sqrt{λ}] × F [\sqrt{λ}].

Therefore, in each case Q_{λ,s} is a quartic étale F -algebra, and the fact that the subalgebra fixed under ν_{λ,s} is F is easily verified.

To prove that Φ is a group homomorphism, consider (λ, s), (λ′, s′) ∈ F^{×} × S^{×} with s = s_{1} + is_{2}, s′ = s′_{1} + is′_{2}. Let d = N_{S/F} (s), d′ = N_{S/F} (s′) and let ω, ξ, η ∈ Q_{λ,s} and ω′, ξ′, η′ ∈ Q_{λ′,s′} be defined as above. The elements

ω_{⋆} = ω ⊗ ω′, ξ_{⋆} = ξ ⊗ ξ′ − η ⊗ η′, η_{⋆} = ξ ⊗ η′ + η ⊗ ξ′

are in (Q_{λ,s} ⊗ Q_{λ′,s′})^{ν_{λ,s}^{−1}⊗ν_{λ′,s′}} and satisfy

ω_{⋆}^{2} = dd′,
ξ_{⋆}^{2} = \frac{λλ′}{2} [dd′ + (s_{1}s′_{1} − s_{2}s′_{2})ω_{⋆}],
ξ_{⋆}η_{⋆} = \frac{λλ′}{2} (s_{1}s′_{2} + s_{2}s′_{1})ω_{⋆},
η_{⋆}^{2} = \frac{λλ′}{2} [dd′ − (s_{1}s′_{1} − s_{2}s′_{2})ω_{⋆}].

Moreover,

(ν_{λ,s} ⊗ Id)(ω_{⋆}) = −ω_{⋆}, (ν_{λ,s} ⊗ Id)(ξ_{⋆}) = η_{⋆}, (ν_{λ,s} ⊗ Id)(η_{⋆}) = −ξ_{⋆}.

Therefore,

(Q_{λ,s}, ν_{λ,s}) ⋆ (Q_{λ′,s′}, ν_{λ′,s′}) ≃ (Q_{λλ′,ss′}, ν_{λλ′,ss′}).

If (λ, s) = (N_{S/F} (z), ℓz^{2}) for some ℓ ∈ F^{×}, z = z_{1} + iz_{2} ∈ S^{×}, then the homomorphism F [W, X, Y] → F^{4} defined by

W ↦ ℓN_{S/F} (z)(1, −1, 1, −1),
X ↦ ℓN_{S/F} (z)(z_{1}, z_{2}, −z_{1}, −z_{2}),
Y ↦ ℓN_{S/F} (z)(z_{2}, −z_{1}, −z_{2}, z_{1})

induces an isomorphism (Q_{λ,s}, ν_{λ,s}) ⟶^{∼} (F^{4}, σ) where σ is the cyclic permutation.

Conversely, suppose that for some (λ, s) ∈ F^{×} × S^{×} there is an isomorphism Q_{λ,s} ≃ F^{4}, and let (ω_{i})_{1≤i≤4}, (ξ_{i})_{1≤i≤4}, (η_{i})_{1≤i≤4} be the images of ω, ξ and η respectively in F^{4}. Then from the relations between ω, ξ and η it follows that z = ω_{1}^{−1}(ξ_{1} + iη_{1}) ∈ S and ℓ = (ξ_{1}^{2} + η_{1}^{2})^{−1}ω_{1}^{3} satisfy

λ = N_{S/F} (z)  and  s = ℓz^{2}.

Therefore, the homomorphism Φ is injective, and it only remains to prove its surjectivity.

Let (Q, ν) be a C_{4}-Galois F -algebra. If Q ≃ F [\sqrt{μ}] × F [\sqrt{μ}] for some μ ∈ F^{×}, then Q ≃ Q_{μ,1}, as was observed at the beginning of the proof. Therefore, for the rest of the proof we may assume Q is a field. Let K = Q^{ν^{2}} ⊂ Q be the subfield fixed under ν^{2}, let K = F (ω) with ω^{2} = d for some d ∈ F^{×}, and let Q = K(ξ) with ξ^{2} = y for some y ∈ K^{×}. We have y ∉ F since Q/F is cyclic. Let y = a + bω with a, b ∈ F, b ≠ 0. Substituting for y an element of the form u^{2}y with u ∈ K, we may assume a ≠ 0. Letting λ = 2ad^{−1} and s_{1} = a^{−1}bd, we may then write y in the form

y = \frac{λ}{2}(d + s_{1}ω).

Let η = ν(ξ). Since ξ^{2} ∈ K and ξ ∉ K, we have ν^{2}(ξ^{2}) = ξ^{2} and ν^{2}(ξ) ≠ ξ, hence ν^{2}(ξ) = −ξ. Therefore, ν(ξη) = −ξη, and it follows that ξη ∈ ωF^{×}. Let

(31)  ξη = \frac{λ}{2}s_{2}ω  for some s_{2} ∈ F^{×}.

From the equation ξ^{2} = \frac{λ}{2}(d + s_{1}ω), we obtain

η^{2} = \frac{λ}{2}(d − s_{1}ω).

Therefore, (31) yields

\frac{λ^{2}}{4} s_{2}^{2}d = \frac{λ^{2}}{4} (d + s_{1}ω)(d − s_{1}ω),

hence d = s_{1}^{2} + s_{2}^{2}. It is then clear that (Q, ν) ≃ (Q_{λ,s}, ν_{λ,s}) with s = s_{1} + is_{2}. □

Corollary 6.3. A quadratic étale F -algebra F [\sqrt{d}] can be embedded in a C_{4}-Galois algebra (Q, ν) as F [\sqrt{d}] ≃ Q^{ν^{2}} if and only if d is a sum of two squares in F.

Proof. The “only if” part was shown in the last lines of the proof of Proposition 6.2. (Alternately, it follows from Propositions 5.10 and 5.11.) The “if” part follows from

the observation that F [\sqrt{d}] ≃ Q_{λ,s}^{ν_{λ,s}^{2}} whenever N_{S/F} (s) = d. □

In other words, this corollary shows that the exact sequence (30) can be extended to

1 → Quad(F) ⟶^{ι^{1}} Cycl_{4}(F) ⟶^{ρ^{1}} Quad(F) ⟶^{δ} Br(F),

where Br(F) is the Brauer group of F and δ maps K = F [\sqrt{d}] to the Brauer class of the quaternion algebra (−1, d)_{F}. Of course, this result is well-known and has an easy cohomological proof.

6.2. Characteristic 2. Cyclic Galois C_{p^{n}}-algebras over fields over characteristic p were constructed by Witt [14, Satz 13], using Witt vectors. The group H^{1}(F, C_{p^{n}}) over a field of characteristic p was computed by Serre in [9, Chap. X, §3], also in terms of Witt vectors. We recall explicitly the results of Serre and Witt for the group C_{4} over a field F of characteristic 2.

Let W_{2}(F) be the additive group of Witt vectors of length 2. By definition we have W_{2}(F) = {(t, s) | t, s ∈ F } with the addition

(t_{1}, s_{1}) .^{+} (t_{2}, s_{2}) = (t_{1} + t_{2}, s_{1} + s_{2} + t_{1}t_{2}).

Alternately,

W_{2}(F) = \{(\begin{matrix}1 & t & s \\ 0 & 1 & t \\ 0 & 0 & 1 \end{matrix})| s, t ∈ F \} ⊂ GL_{2}(F).

The neutral element is (0, 0) and . − (t, s) = (t, s + t^{2}). The map

℘_{2} : W_{2}(F) → W_{2}(F), (t, s) ↦ (t^{2}, s^{2}) . − (t, s) = (t^{2} + t, s^{2} + s + t^{2} + t^{3})

is a group homomorphism and there is an exact sequence of Γ-modules:

(32)  0 → C_{4} → W_{2}(F_{s}) ⟶^{℘_{2}} W_{2}(F_{s}) → 0,

where C_{4} is identified with the subgroup of W_{2}(F_{s}) generated by (1, 0).

Proposition 6.4. H^{1}(Γ, W_{2}(F_{s})) = 0 and H^{1}(Γ, C_{4}) ≃ W_{2}(F)/℘_{2}(W_{2}(F)).

Proof. The first claim follows from the exact sequence of (additive) Γ-modules:

0 → F_{s} →^{ι} W_{2}(F_{s}) →^{π} F_{s} → 0

where ι(s) = (0, s) and π(t, s) = t, and the fact that H^{1}(Γ, F_{s}) = 0 (by the additive version of Hilbert’s Theorem 90). The last claim follows from the exactness of (32) and from the first claim. □

For (t, s) ∈ W_{2}(F), let I_{(t,s)} ⊂ F [W, X] be the ideal generated by the polynomials f_{1}(W), f_{2}(W, X) such that

(f_{1}(W), f_{2}(W, X)) = ℘_{2}(W, X) . − (t, s),

and let

Q_{(t,s)} = F [W, X]/I_{(t,s)}.

Letting ω, ξ be the images of W, X in Q_{(t,s)}, the relations defining Q_{(t,s)} can be rewritten as

ω^{2} + ω = t, ξ^{2} + ξ = tω + s.

It is therefore clear that Q_{(t,s)} is a quartic 2-algebra over F. The automorphism of F [W, X] given by

(W, X) ↦ (W, X) .^{+} (1, 0)

induces an automorphism ν_{(t,s)} of Q_{(t,s)}; we have by definition ν_{(t,s)}(ω) = ω + 1 and ν_{(t,s)}(ξ) = ξ + ω, hence the fixed subalgebra of Q_{(t,s)} is F. Thus, (Q_{(t,s)}, ν_{(t,s)}) is a C_{4}-Galois F -algebra.

Proposition 6.5. The map (t, s) ↦ (Q_{(t,s)}, ν_{(t,s)}) induces a group isomorphism

Ψ: W_{2}(F)/℘_{2}(W_{2}(F)) ⟶^{∼} Cycl_{4}(F).

Proof. Let (t, s), (t′, s′) ∈ W_{2}(F), and let ω, ξ ∈ Q_{(s,t)} and ω′, ξ′ ∈ Q_{(s′,t′)} be the elements defined as above. In Q_{(t,s)} ⊗ Q_{(t′,s′)}, consider the elements

ω_{⋆} = ω ⊗ 1 + 1 ⊗ ω′, ξ_{⋆} = ξ ⊗ 1 + 1 ⊗ ξ′ + ω ⊗ ω′.

Computation shows that ω_{⋆} and ξ_{⋆} are invariant under ν_{(t,s)}^{−1} ⊗ ν_{(t′,s′)} and satisfy

ω_{⋆}^{2} + ω_{⋆} = t + t′, ξ_{⋆}^{2} + ξ_{⋆} = (t + t′)ω_{⋆} + s + s′ + tt′.

Moreover, (ν_{(t,s)} ⊗ Id)(ω_{⋆}) = ω_{⋆} + 1 and (ν_{(t,s)} ⊗ Id)(ξ_{⋆}) = ξ_{⋆} + ω_{⋆}. Therefore,

(Q_{(t,s)}, ν_{(t,s)}) ⋆ (Q_{(t′,s′)}, ν_{(t′,s′)}) ≃ (Q_{(t,s) .^{+} (t′,s′)}, ν_{(t,s) .^{+} (t′,s′)}).

If (t, s) = ℘_{2}(x, y) = (x^{2} + x, y^{2} + y + x^{3} + x^{2}) for some x, y ∈ F, then the map F [W, X] → F defined by

W ↦ (x, x + 1, x, x + 1), X ↦ (y, x + y, y + 1, x + y + 1)

induces an isomorphism (Q_{(t,s)}, ν_{(t,s)}) ⟶^{∼} (F^{4}, σ) where σ is the cyclic permutation of factors.

Conversely, if Q_{(t,s)} ≃ F^{4} for some t, s ∈ F, then letting (ω_{i})_{1≤i≤4} and (ξ_{i})_{1≤i≤4} denote the images of ω and ξ in F^{4}, it is readily verified that

t = ω_{1}^{2} + ω_{1}  and  s = ξ_{1}^{2} + ξ_{1} + ω_{1}^{3} + ω_{1}^{2},

hence (t, s) = ℘_{2}(ω_{1}, ξ_{1}). This shows that the map Ψ is an injective homomorphism of groups, and it only remains to prove its surjectivity.

Let (Q, ν) be a C_{4}-Galois algebra. If Q ≃ F [℘^{−1}(u)]×F [℘^{−1}(u)] for some u ∈ F, then (Q, ν) ≃ (Q_{(0,u)}, ν_{(0,u)}). For the rest of the proof, we may thus assume Q is a field. Let K = Q^{ν^{2}} ⊂ Q be the subfield fixed under ν^{2} and let Q = K(ξ) with ξ^{2} + ξ ∈ K. Then ν^{2}(ξ) = ξ + 1, and it follows that the element ω = ξ + ν(ξ) satisfies

ν^{2}(ω), ν(ω) = ω + 1,

hence K = F (ω). We then have ξ^{2} + ξ = tω + s for some t, s ∈ F, and

ω^{2} + ω = (ξ^{2} + ξ) + ν(ξ^{2} + ξ) = t.

Therefore, (Q, ν) ≃ (Q_{(t,s)}, ν_{(t,s)}). □

Corollary 6.6. Every quadratic étale F -algebra can be embedded in a C_{4}-Galois F -algebra (Q, ν) as Q^{ν^{2}} ≃ Δ(Q).

Proof. For any t ∈ F, we have F [℘^{−1}(t)] ≃ Q_{(t,s)}^{ν_{(t,s)}^{2}} for all s ∈ F. □

The corollary shows that the last map in the exact sequence (30) is onto when char F = 2. This is clear from a cohomological viewpoint, since the cohomological 2-dimension of Γ is at most 1, see [11, p. 86].

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Departement Mathematik, ETH Zentrum, CH-8092 Zürich, Switzerland

E-mail address: knus@math.ethz.ch

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@math.ucl.ac.be