On the annihilating ideal for trace forms

Martin Epkenhans

Abstract. We give several examples of classes of trace forms for which the ideal of annihilating polynomials is principal. We prove, that in general, the annihilating ideal is not a principal ideal.

2000 Mathematics Subject Classification: 11E81, 12F10 Keywords and Phrases: Quadratic form, trace form, annihilating polynomial, Burnside ring

1. Introduction

Let K be a field of characteristic different from 2. Since the Witt ring W (K) over K is an integral ring we may consider polynomials in ℤ[X] evaluated at an element ϕ of W (K). We say a polynomial p(X) ∈ ℤ[X] annihilates ϕ if p(ϕ) = 0 in W (K).

Definition 1. Let M be any class of quadratic forms. Then the annihilating ideal I_{M} of M is defined to be

I_{M} := {f(X) ∈ ℤ[X] | f(ϕ) = 0 ∈ W (K) for all ϕ ∈ M}.

During the last 15 years several examples of annihilating polynomials of quadratic forms have appeared in the literature. Let us first recall some of these results and present them in the context of annihilating ideals. D. Lewis [11] gives an annihilating polynomial for quadratic forms of dimension n.

Theorem 1 (Lewis). Let Q_{n} be the class of all quadratic forms of dimension n. Then I_{Q_{n}} is a principal ideal generated by the Lewis polynomial

p_{n}(X):=\{ \begin{matrix}X(X^{2} − 2^{2})(X^{2} − 4^{2}) ⋯ (X^{2} − n^{2}), & if n ≡ 0 mod 2, \\ (X^{2} − 1^{2})(X^{2} − 3^{2}) ⋯ (X^{2} − n^{2}), & if n ≡ 1 mod 2. \end{matrix}

As usual, (a) = aR denotes the principal ideal generated by the element a ∈ R.

Example 1. Let P_{n} be the class of all n-fold Pfister forms. Then I_{P_{n}} = (X^{2} − 2^{n}X).

Now let us recall the definition of the trace form. Let A be a finite dimensional étale K-algebra. With it we associate the quadratic form

< A/K >: A → K : x ↦ trace_{A/K}(x^{2}),

which is called the trace form of A/K. From some unpublished result of P.E. Conner [2] we get the following theorem.

Theorem 2 (Conner). Let T_{n} be the class of trace forms of all separable field extensions of degree n of fields of characteristic ≠ 2. Then I_{T_{n}} = (C_{n}(X)), where C_{n}(X) = ∏_{k≥0,p_{n}(k)=0}(X − k).

In a certain sense, the result of P.E. Conner has been improved first by P. Beaulieu and T. Palfrey [1] and later on by D. Lewis and S. McGarraghy [12]. Consider a separable field extension L/K of degree n and let N/K be a normal closure of L/K. Let f(X) ∈ K[X] be the minimal polynomial of a primitive element α of L/K. Then Beaulieu and Palfrey introduced the notion of the Galois number of a polynomial f(X). This number is defined to be the smallest number g_{f} such that any g_{f} roots of f(X) generate a splitting field of f(X). In a group theoretical context the Galois number of a G-set is defined to be the smallest natural number G such that any group element σ ∈ G with g fixed points acts as the identity. This number is also called the minimal degree of a permutation representation. See [9] for the determination of Galois numbers of doubly transitive groups. Now the polynomial of Beaulieu and Palfrey is defined to be

B_{f}(X) := (X − n) · ∏_{k=0,k≡nmod2}^{g−1}(X − k).

Theorem 3 (Beaulieu-Palfrey). The polynomial B_{f}(X) has the property that B_{f}(ϕ) = 0 ∈ W (K), where ϕ is isometric to the trace form of the field extension given by the separable and irreducible polynomial f(X) ∈ K[X].

Denote the Galois group of N/K by G(N/K). Then the action of G(N/K) on the left cosets of G(N/K)/G(N/L) defines a Gset. For any G-set S of cardinality n and any subgroup U < G let inv_{U}(S) := ♯{s ∈ S | s^{σ} = s for all σ ∈ U} be the number of fixed points of the restricted action. Set inv(S) := {inv_{U}(S) | U < G}. The definition of the following polynomial is due to Lewis and McGarraghy (see [12] corollary 3.5). For any G-set S set

φ(S) := {inv_{U}(S) | U < G, inv_{U}(S) ≡ n mod 2}.

Now set

p_{G,S} := ∏_{k∈φ(S)}(X − k).

Theorem 4 (Lewis-McGarraghy). Let L/K be a finite and separable field extension and let p_{G,S}(X) be defined as above. Then p_{G,S}(X) annihilates the trace form of L/K.

Note, that the result above can be generalized to trace forms of étale algebras. Using Springer’s theorem on the lifting of quadratic forms according to odd degree extensions, we get annihilating polynomials of lower degree (see [8][Theorem 2.10]). Now we come to the definition of the class of quadratic forms we like to discuss in this paper.

Definition 2. Let G be a finite group and let H < G be a subgroup with ∩_{σ∈G}σHσ^{−1} = 1. Then the class M(G, H) consists of those quadratic forms ϕ such that

1. there exists an irreducible and separable polynomial f(X) ∈ K[X] with Galois group Gal(f) isomorphic to G;

2. the action of Gal(f) on the roots of f(X) and the action of G on the left cosets G/H are equivalent;

3. ϕ and the trace form < (K[X](f(X)))/K > are isometric.

Note, that the condition on H guarantees, that G acts faithfully on G/H. The work of Lewis [11] and Conner [2] give annihilating polynomials for quadratic forms, resp. trace forms of dimension n.

To finish the determination of the annihilating ideal, we have to consider signatures. The zeros of p_{n}(X), resp. C_{n}(X) are exactly those integers, which occur as signature values of quadratic forms in Q_{n}, resp. T_{n}.

Definition 3. Let M be a class of quadratic forms. Then the set of signatures of M is denoted

sign(M) := {s ∈ ℤ | s = sign ϕ for some ϕ ∈ M}.

If sign(M) is finite, the signature polynomial of M is given by

Sign_{M}(X) := ∏_{k∈sign(M)}(X − k).

Since the signature is a ring homomorphism we get

Proposition 1. Let M be a class of quadratic forms. If sign(M) is a finite set, then

I_{M} ⊂ (Sign_{M}(X)).

Otherwise, we get I_{M} = (0).

The signature of a trace form < L/K > equals the number of real embeddings of L into ℝ, which is the number of real roots of a polynomial f(X) with L ≃ K[X]/(f(X)) [13]. This observation gives rise to the definition of a signature in a group theoretical setting.

Definition 4. Let S be a finite G-set and σ ∈ G with σ^{2} = 1. Then

sign_{σ}(S) := ♯{s ∈ S | s^{σ} = s}

is called the signature of S according to σ.

sign(S) := {sign_{σ}(S) | σ ∈ G, σ^{2} = 1}

is the set of signatures of S.

Observe, that sign_{σ}(S) = inv_{<σ>}(S). From proposition 3 in [6] we conclude

Proposition 2. Let G be a finite group and let H be a subgroup of G with ∩_{σ∈G}σHσ^{−1} = 1. Then

Sign(M(G, H)) = Sign(G/H).

2. Burnside rings

The proofs of Conner, Beaulieu-Palfrey and Lewis-McGarraghy are based on certain identities in the Burnside ring B(G) of G and translated into identities in the Witt ring by applying a homomorphism given by Dress [5] (see also [7] proposition 3). Let B(G) denote the Burnside ring of G-sets (for more details see [4][chapter 11 §80]). Let χ_{H}^{G} denote the G-set given by the action of G on the set of left cosets G/H. Let S = S(G) be a full set of nonconjugate subgroups of G. By corollary 80.6 in [4]

B(G) = ⊕_{H∈S}ℤ · χ_{H}^{G}.

For any σ ∈ G, σ^{2} = 1, the definition of signatures given in definition 4 gives rise to a signature homomorphism

sign_{σ} : B(G) → ℤ.

Let L(G) := ∩_{σ∈G,σ^{2}=1}ker(sign_{σ}) be the kernel of the total signature homomorphism. For any Galois extension N/K with Galois group G(N/K) isomorphic to G there is a ring homomorphism h_{N/K} : G → W (K). Let T (G) := ∩ker(h_{N/K}) denote the trace ideal of G (see [6],[7],[8] for more details). Here N/K runs over all Galois extensions of fields of characteristic ≠ 2 with Galois group G(N/K) ≃ G. Then T (G) ⊂ L(G). Theorem 16 in [7] states

Theorem 5. Let G be a finite group. Then L(G)/T (G) is a finite 2-group.

(B(G)/T (G))_{tor} = L(G)/T (G)

and the only torsions in B(G)/T (G) are 2-torsions.

Together with proposition 1 we conclude

Corollary 1. Let G be a finite group and let H be a subgroup with ∩_{σ∈G}σHσ^{−1} = 1. Then there is an integer l ∈ ℕ_{0} such that

(2^{l}q_{G,H}(X)) ⊂ I_{M(G,H)} ⊂ (q_{G,H}(X)).

We can choose 2^{l} to be the exponent of the finite abelian 2-group L(G)/T (G).

Let (𝔞 : 𝔟) := {x ∈ R | x𝔟 ⊂ 𝔞} be the ideal quotient of the ideals 𝔞, 𝔟 in the ring R. Then

I_{M(G,H)} = (I_{M(G,H)} : (q_{G,H}(X))) · (q_{G,H}(X)).

Since I_{M(G,H)} contains monic polynomials, we get

Corollary 2. I_{M(G,H)} is principal ideal if and only if I_{M(G,H)} = (q_{G,H}(X)).

We introduce some more notations. For any subgroup H of G let res_{H}^{G} : B(G) → B(H) denote the restriction homomorphism (see [7]). Let < a_{1}, …, a_{n} > denote the diagonal matrix with diagonal entries a_{1}, …, a_{n}. For n ∈ ℕ and a matrix A denote the n-fold orthogonal sum by n × A := ⊕_{i=1}^{n}A.

Proposition 3. Let e(G) denote the exponent of L(G)/T (G). If any subgroup of a 2-Sylow subgroup G_{2} of G is a normal subgroup in G_{2}, then

(e(G)/2) · q_{G,H} ∈ I_{M(G,H)}.

Proof. The proof of proposition 4.3 in [8] implies res_{G_{2}}^{G}(q_{G,H}(χ_{H}^{G})) ∈ 2 · B(G). Since q_{G,H}(χ_{H}^{G}) ∈ L(G) we are done by corollary 1. □

The following theorem gives an affirmative answer to a question asked in [7]. With it we are able to translate certain problems on annihilating polynomials to the corresponding problems over 2groups.

Theorem 6. Let G be a finite group with 2-Sylow subgroup G_{2}. Then for any element χ ∈ B(G) we get

χ ∈ T (G) ⇔ res_{G_{2}}^{G}(χ) ∈ T (G_{2}).

Hence the restriction homomorphism induces an injection

L(G)/T (G) ↪ L(G_{2})/T (G_{2}).

Proof. From lemma 4.2a in [6] we know res_{G_{2}}^{G}(χ) ∈ T (G_{2}) implies χ ∈ T (G). Let n := ord(G) and let N/K be a Galois extension with Galois group G(N/K) ≃ G_{2}. Set L := K(X_{1}, …, X_{n}), where X_{1}, …, X_{n} are algebraically independent indeterminates. The regular representation of G defines a monomorphism G ↪ 𝔖({X_{1}, …, X_{n}}), where 𝔖({X_{1}, …, X_{n}}) denotes the symmetric group of the set {X_{1}, …, X_{n}}. Hence G is a subgroup of the group of automorphisms Aut_{K}(L). Set F := L^{G} and F_{2} := L^{G_{2}}. Since G acts transitively on {X_{1}, …, X_{n}}, the polynomial f(X) := (X − X_{1}) ⋯ (X − X_{n}) is irreducible over F. Hence X_{1} is a primitive element of L/F and of L/F_{2}. Assume, that X_{1}, …, X_{m}, m := ord(G_{2}) are the conjugates of X_{1} over F_{2}. For any H < G_{2} we can choose a primitive element α_{H} of L^{H}/F_{2} of the form ∑_{i=0}^{m−1} g_{i}X_{1}^{i} with g_{i} ∈ L[X_{1}, …, X_{n}] ∩ F_{2} =: R. Let χ ∈ T (G). Then h_{L/F} (χ) = 0 implies res_{G_{2}}^{G}(χ) = ∑_{H∈S} m_{H}χ_{H} ∈ ker(h_{L/F_{2}}), where H runs over a full set S of nonconjugate subgroups of G_{2}. For any H ∈ S calculate a matrix M_{H} of < N^{H}/F_{2} > with respect to the F_{2}-basis 1, α_{H}, …, α_{H}^{[G_{2}:H]−1}. Hence M_{H} ∈ Gl(n_{H}, R) with n_{H} := [G_{2} : H]. Since h_{L/F_{2}}(res_{G_{2}}^{G}(χ)) = 0 ∈ W (F_{2}), there is a matrix A ∈ Gl(t, R) and a non-zero polynomial g ∈ R with

A · [⊕_{H∈S}m_{H}M_{H}] · A^{T} = g^{2} · (t/2× < 1, −1 >).

Let α ∈ N be a primitive element of N/K and let α_{1} := α, α_{2}, …, α_{m} be the conjugates of α over N^{G_{2}}. Label these elements according to the action of G_{2} on {X_{1}, …, X_{m}}. Now we choose algebraically independent indeterminates Y_{1}, …, Y_{m} and set Z_{1} := Y_{1}+α_{1}Y_{2}+α_{1}^{2}Y_{3}+…+α_{1}^{m−1}Y_{m}, …, Z_{m} := Y_{1} + α_{m}Y_{2} + α_{m}^{2}Y_{3} + … + α_{m}^{m−1}Y_{m}. By looking at the Vandermonde determinant we see that Z_{1}, …, Z_{m} are algebraically independent. Replace X_{1}, …, X_{m} by Z_{1}, …, Z_{m}. Denote the new polynomials resp. matrices by \overline{g}, resp. \overline{M}_{H}. Hence

\overline{A} · [⊕_{H∈S}m_{H} \overline{M}_{H}] · \overline{A}^{T} = \overline{g}^{2} · (t/2× < 1, −1 >),

and \overline{g} ≠ 0. Since the set of primitive elements of a separable field extension is a non-empty Zariski-open subset, there is an n-tuple a = (a_{1}, …, a_{n}) ∈ K^{n}, such that \overline{g}(a_{1}, …, a_{n}) ≠ 0 and for any H ∈ S the element ∑_{i=0}^{m−1} \overline{g}_{i}(a)α_{1}^{i} is a primitive element of N^{H}/K. We get

\overline{A}(a) · [⊕_{H∈S}m_{H} \overline{M}_{H}(a)] · \overline{A}(a)^{T} = \overline{g}(a)^{2} · (t/2× < 1, −1 >),

and \overline{g}(a) ∈ K, \overline{A}(a) ∈ M(t, K), \overline{M}_{H}(a) ∈ M(n_{H}, K). Hence h_{N/K}(res_{G_{2}}^{G}(χ)) = 0 ∈ W (K). □

3. Groups with quaternion 2-Sylow subgroups

This section contains a class of examples, where I_{M(G,H)} is not a principal ideal.

Proposition 4. Let G be a finite group with 2-Sylow subgroup a quaternion group of order 8. Let H < G be a subgroup of G with ∩_{σ∈G}σHσ^{−1} = 1. Then

1.

I_{M(G,H)} = (q_{G,H}(X)), if

(a) ord(H) ≡ 1 mod 2,

(b) ord(H) ≡ 2 mod 4,

(c) ord(H) ≡ 0 mod 4 and the permutation representation of

G on G/H contains only even permutations.

2. Let ord(H) ≡ 0 mod 4 and suppose, that the permutation representation of G on G/H contains an odd permutation.

(a) Then G is a semidirect product of G_{2} and a normal subgroup A of odd order. The conjugation of G_{2} on A induces a monomorphism Φ : G_{2} ↪ Aut(A).

(b) Let n := [G : H] and s := sign_{σ}χ_{H} with σ ∈ G_{2} the unique involution of G_{2}. Then

I_{M(G,H)} = \{ \begin{matrix}(X, 2) · ((X − n)(X − s)), & if n ≡ 4 mod 8, \\ (X − 1, 2) · ((X − n)(X − s)), & if n ≡ 0 mod 8. \end{matrix}

Proof. 1) see proposition 5.2 in [8]. 2(a) follows from lemma 5.2 in [8]. 2(b): We use the notation of §5 in [8]. By proposition 6 and [6] proposition 7 we get 2(X − n)(X − s) ∈ I_{M(G,H)}. Assertion. X(X − n)(X − s) ∈ I_{M(G,H)} if 8 ∤ n. By proposition 7 in [6] we have to determine the coefficients m_{i} of χ_{H_{i}} in res_{G_{2}}^{G}(χ_{H}^{G}(χ_{H}^{G} − nχ_{G}^{G})(χ_{H}^{G} − sχ_{G}^{G})). Since a = 0 we get m_{i} = a′b_{i} + 2b_{i}^{2}(2b_{i} − n − s). Observe, that a′ = ns ≡ 0 mod 4 and n + s ≡ 2a ≡ 0 mod 4. Hence m_{i} ≡ 0 mod 4. We conclude (2, X) ⊂ (I_{M(G,H)} : (q_{G,H}(X))). By the preceeding theorem and by proposition 5.5 in [8] q_{G,H}(X) ∉ I_{M(G,H)}. Since (2, X) is a maximal ideal in ℤ[X], we are done. The case 8 | n is left to the reader. □

The smallest example is as follows. The automorphism group of A = ℤ/3ℤ × ℤ/3ℤ is a double cover of 𝔖_{4}, which contains Q_{8} as a subgroup. Set G := A ⋊ Q_{8} and let H be a subgroup of order 4 in G. Then I_{M(G,H)} = (X, 2) · ((X − 18)(X − 2)) (see [8][Example 5.6]).

4. Some more 2-groups

Lemma 1. Let G be a finite 2-group and let H =< τ > be a subgroup of order 2 in G with ∩_{σ∈G}σHσ^{−1} = 1. Then

q_{G,H} = X(X − sign_{τ}χ_{H})(X − ord(G)/2);
sign_{τ}χ_{H} = ord(C_{G}(τ))/2;
I_{M(G,H)} = (q_{G,H}).

Here C_{G}(τ) denotes the centralizer of τ in G.

Proof. Set n := [G : H] = ord(G)/2. Since τ is not contained in the non-trivial center of G, we get X | q_{G,H}(X) by lemma 3.3(2) in [8]. Now proposition 10 and corollary 11 in [7] gives the result on q_{G,H}(X) and the signature value. We easily calculate χ_{H}^{2} = sign_{τ}(χ_{H}) · χ_{H} + \frac{n−sign_{τ} χ_{H}}{2} · χ_{1}. Hence

q_{G,H}(χ_{H}) = (χ_{H}^{2} − sign_{τ}(χ_{H}) · χ_{H})(χ_{H} − nχ_{G})
= \frac{n − sign_{τ}χ_{H}}{2} · χ_{1} · (χ_{H} − nχ_{G})
= \frac{n − sign_{τ}χ_{H}}{2} · (nχ_{1} − nχ_{1}) = 0.

□

5. Examples

Finally, let us summarize some examples, where I_{M(G,H)} is a principal ideal.

Theorem 7. In the following cases we get

I_{M(G,H)} = (q_{G,H}(X))

1. G has odd order. Then I_{M(G,H)} = (X − n).

2. G_{2} is elementary abelian or cyclic.

3. G_{2} is a dihedral group of order 2^{m} ≥ 8.

4. H = 1.

5. G is abelian.

6. G is a Frobenius group.

7. G is a Zassenhaus group ≠ P ML(2, q).

8. G = ^{2}G_{2}(q), q = 3^{2m+1}, m ≥ 1 the Ree group in its doubly transitive permutation representation of degree q^{3} + 1 and H a one point stabilizer.

9. G a group of order ≤ 31.

10. G = Q_{2^{l}}, M(2^{l}), QD_{2^{l}}.

There exists four groups of order 2^{l+1} ≥ 8, which contain an element of order 2^{l}. Beside the dihedral group D_{2^{l}} there are the generalized quaternion group Q_{2^{l}}, the quasidihedral group QD_{2^{l}} and the group M(2^{l}) (see [10][I. Satz 14.9]).

Proof. For (1) see [3] corollary 1.6.5, resp. proposition 17 in [7]. Use proposition 3 and [6] proposition 5 and 6 to prove (2).

(3) follows from proposition 5.1 in [8].

4) If H = 1, then q_{G,H}(X) = B_{G,H}(X) = X −n, resp. = X(X −n).

5) The condition on H implies H = 1. (6), (7) and (8) follow from [8] proposition 6.1, 6.3 and 3.4.

9) By (1), (2), (3) and (5) it remains to consider non-abelian groups of order n = 8, 16, 24 and the case n = 24, G_{2} ≃ ℤ/2ℤ × ℤ/4ℤ. n = 8. Apply (3), resp. (5) in the case of the quaternion group. n = 16. Any subgroup H < G of order ≥ 4 has a non-trivial intersection with the center of G (see [14]). Now use lemma 1. n = 24. The result for G_{2} ≃ ℤ/2ℤ × ℤ/4ℤ follows from some unpublished determination of the exponent of L(G)/T (G) for G = ℤ/2ℤ × ℤ/2^{l}ℤ. Since there is no injection Q_{8} ↪ Aut(ℤ/3ℤ), we are done by proposition 4.

(10) follows from lemma 1, since any subgroup H of G of order ≥ 4 has a non-trivial intersection with the center of G. □

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Fb Mathematik

Universität Paderborn D-33095 Paderborn martine@unipaderborn.de