NONTRIVIALITY OF NK_{1}(D) FOR DIVISION ALGEBRAS

ROOZBEH HAZRAT AND ADRIAN WADSWORTH

Abstract. A field E is said to be NKNT if for any noncommutative division algebra D finite dimensional over E ⊆ Z(D) = F with index ind(D), Nrd(D^{∗})/F^{∗ind(D)} is nontrivial. It is proved that if E is a field finitely generated but not algebraic over some subfield then E is NKNT. As a consequence, if F = Z(D) is finitely generated over its prime subfield or over an algebraically closed field, then CK_{1}(D) = Coker(K_{1}F → K_{1}D) is nontrivial.

Let D be a division algebra over its center F of index n. Denote by D^{∗} and F^{∗} the multiplicative group of D and F respectively. Let Nrd_{D}: D^{∗} → F^{∗} be the reduced norm map, D^{(1)} the kernel of this map and D′ the commutator subgroup of D^{∗}. The inclusion map F ↪ D induces a homomorphism K_{1}(F) = F^{∗} → K_{1}(D) = D^{∗}/D′. Consider the group

CK_{1}(D) = Coker(K_{1}F → K_{1}D) ≅ D^{∗}/F^{∗}D′ .

Since x^{−n}Nrd(x) ∈ D^{(1)} and the reduced Whitehead group SK_{1}(D) = D^{(1)}/D′ is n-torsion (by [3], p. 157, Lemma 2), it follows that CK_{1}(D) is an abelian group of bounded exponent n^{2} (In fact one can show that the bound is n, see the proof of Lemma 4, p. 154 in [3] or pp. 579–580 in [5]). Thus, by the Prüfer-Baer theorem CK_{1}(D) ≅ ⨁ ℤ_{k_{i}} where each k_{i} | n (see [11], p. 105). Therefore if CK_{1}(D) is nontrivial then D^{∗} has a (normal) maximal subgroup. The question of whether D^{∗} has a maximal subgroup seems to remain open and thus by the above observation is limited to the case when CK_{1}(D) is trivial. In [4], Th. 2.12 it was proved that if D is a tensor product of cyclic algebras then CK_{1}(D) is trivial if and only if D is a quaternion division algebra (\frac{−1,−1}{F}) where F is a real Pythagorean field (see also [8]). It has been conjectured in [6] that if CK_{1}(D) is trivial then D is a quaternion division algebra.

The group CK_{1} has been computed in [5] for certain division algebras, and its connection with SK_{1} was also studied. But, CK_{1}(D) is often difficult to work with. We will focus here on a related invariant, NK_{1}(D), which is sometimes more tractable, and can yield information about CK_{1}(D). Define

NK_{1}(D) = D^{∗}/F^{∗}D^{(1)} ≅ Nrd(D^{∗})/F^{∗ind(D)}

(with the isomorphism given by the reduced norm map). Observe that NK_{1}(D) is a homomorphic image of CK_{1}(D) and that whenever SK_{1}(D) = 1, we have CK_{1}(D) = NK_{1}(D). (Recall that SK_{1}(D) = 1 whenever ind(D) is square-free, or the center F of D is a local or a global field, by [3], p. 164, Cor. 4, Th. 3, p. 165, (17), p. 166, (18).) For example, if Q = (\frac{a,b}{F}) is a quaternion division algebra with char(F) ≠ 2, we have

Date: 31 jan 06.

CK_{1}(Q) = NK_{1}(Q) ≅ ({r^{2} − as^{2} − bt^{2} + abu^{2} | r, s, t, u ∈ F }\{0})/F^{∗2}.

From this formula, it is immediate that CK_{1}(Q) is trivial iff F is a real Pythagorean field and Q ≅ (\frac{−1,−1}{F}).

Observe that the condition that NK_{1}(D) be trivial for a noncommutative division ring D is an extremely strong one. Indeed, if ind(D) = d then NK_{1}(D) = 1 iff Nrd_{D}(D^{∗}) = F^{∗d} = Nrd_{D}(F^{∗}), which holds iff for every maximal subfield L of F, N_{L/F} (L^{∗}) = F^{∗d} = N_{L/F} (F^{∗}). It was shown in [4] that if C is a noncommutative cyclic algebra with NK_{1}(C) = 1, then C ≅ (\frac{−1,−1}{F}) with F a Pythagorean field. We will show here that NK_{1}(D) is nontrivial for a great many other noncommutative division algebras D. Of course, whenever NK_{1}(D) ≠ 1, we also have CK_{1}(D) ≠ 1.

Definition. A field E is said to be NKNT (for NK_{1} nontrivial) if for any noncommutative division algebra D finite dimensional over E (and not necessarily central over E), NK_{1}(D) is nontrivial.

It is clear from the definition that if a field E is NKNT then so is every finite degree field extension of E. Here are some examples of NKNT fields: Clearly a finite field is NKNT; so is any algebraically closed field; so also is any field of transcendence degree 1 over an algebraically closed field, by Tsen’s Theorem. Since every division algebra over a global field is a cyclic algebra, the result quoted above shows that every global field is NKNT. Likewise, every nonreal local field is NKNT. However the field of real numbers ℝ is not NKNT, since ℝ is a real Pythagorean and thus CK_{1}(ℍ_{ℝ}) = 1, but the rational function field ℝ(t) is NKNT. This is a consequence of our main theorem below. But we can see it directly as follows: If L is a finite degree extension of ℝ(t) and D is an L-central noncommutative division algebra, then by Tsen’s Theorem, D is split by L(\sqrt{−1}), so D is a quaternion algebra; but NK_{1}(D) is then nontrivial because L is not Pythagorean.

In this note our main result is:

Theorem. Let F be a field which is finitely generated but not algebraic over some subfield F_{0}. Then, F is NKNT.

Let D be a division algebra with center F. In showing that NK_{1}(D) is nontrivial, Lemma 1 below allows us to reduce to the case where ind(D) is a prime power. Our arguments then divide into two cases depending on whether ind(D) is a power of char(F). Lemma 2 and its Corollary handle the first case:

Lemma 1. Let D_{1}, …, D_{k} be division algebras with center F such that gcd(ind(D_{i}), ind(D_{j})) = 1 whenever i ≠ j.Then,

NK_{1}(D_{1} ⊗_{F} D_{2} ⊗_{F} … ⊗_{F} D_{k}) ≅ NK_{1}(D_{1}) × … × NK_{1}(D_{k}) .

NONTRIVIALITY OF NK_{1}(D) 3

Proof. It suffices by induction to prove the result for k = 2. This can be done the same way as the corresponding result for CK_{1} was proved in [5], Th. 2.8. □

Lemma 2. Let D be a noncommutative division algebra similar to a cyclic algebra A. In each of the following cases NK_{1}(D) is nontrivial:

(1) F contains a square root of −1;

(2) The characteristic of F is 2;

(3) The degree of A is odd.

Proof. Since the primary components of D are similar to tensor powers of A which are similar to cyclic algebras, it suffices by Lemma 1 to consider the case when ind(D) is a power of a prime. Thus, assume ind(D) = p^{c}, where p is a prime number and c ≥ 1 and D is similar to a cyclic algebra A = (E/F, σ, a). Choose A of minimal degree. Then, deg(A) = p^{e} and a ∉ F^{∗p}; for, if a = b^{p}, then A is Brauer equivalent to (E_{0}/F, σ, b), where [E : E_{0}] = p, contradicting the minimality of deg(A).

Let d = p^{e} = deg(A). Let α be the standard generator of A with α^{d} = a. Since the powers of α up to the d-th are part of a base of A over F, they are F -linearly independent. Therefore, the minimal polynomial of α over F is x^{d} − a. If M is any splitting field of A, then for α⊗1 ∈ A⊗_{F} M, the minimal polynomial of α⊗1 over M is again x^{d} − a. Since the characteristic polynomial of α⊗1 has degree d, this polynomial is also x^{d} − a. Hence, Nrd_{A}(α) = det(α⊗1) = (−1)^{d−1}a. So, if p is odd, or char(F) = 2, or F contains a square root of −1, then Nrd_{A}(α) ∉ F^{∗p}. But thanks to the Dieudonné determinant, Nrd_{A}(A^{∗}) = Nrd_{D}(D^{∗}). Thus Nrd_{A}(α) ∈ Nrd_{D}(D^{∗})\F^{∗p^{e}}, so NK_{1}(D) is nontrivial. □

Recall that a p-algebra is a central simple algebra of degree a power of the prime p over a field of characteristic p. Albert’s main theorem in the theory of p-algebras states that every p-algebra is similar to a cyclic p-algebra (see [1], p. 109, Th. 31). Combining this with the Lemma above, we obtain:

Corollary 3. Let D be a noncommutative p-division algebra. Then NK_{1}(D) is nontrivial.

Remark. Let G(D) = D^{∗}/Nrd(D^{∗})D′, which is a bigger group than CK_{1}(D) in general. It is much easier to see that G(D) ≠ 1 for every noncommutative p-division algebra D. Indeed, if G(D) = 1 then Nrd(D^{∗}) = Nrd(D)^{∗p^{n}} where ind(D) = p^{n}. So, for F = Z(D),

F^{∗p^{n}} ⊆ Nrd(D^{∗}) = Nrd(D)^{∗p^{n}} = Nrd(D)^{∗p^{2n}} ⊆ F^{∗p^{2n}}.

Hence, F^{∗}^{p^{n}} = F^{∗}^{p^{2n}}. Since char(F) = p, this implies F^{∗} = F^{∗}^{p}, i.e., F has no proper purely inseparable extensions. But one knows by [1], p. 104, Th. 21, that any p-algebra has a purely inseparable splitting field; hence, D = F, a contradiction. (Compare this argument with [8], Th. 2).

In order to prove the main Theorem, we need two propositions.

Proposition 4. Let F ⊆ L be fields with [L:F] = d < ∞ such that N_{L/F} (L^{∗}) = F^{∗d}. If V is a discrete valuation ring of F with residue field \overline{V}, and if the integral closure of V in L is a finite V -module, then V has a unique extension to a DVR W of L, and [\overline{W}:\overline{V}] = [L:F].

Proof. Let v be the normalized discrete valuation on F corresponding to the valuation ring V. “Normalized” means that the value group v(F^{∗}) = ℤ. Let v_{1}, …, v_{s} be all the (inequivalent) extensions of v to L. Since v is discrete and the integral closure of V in L is a finite V -module, we have ∑_{i=1}^{s} e_{i}f_{i} = [L:F], where e_{i} is the ramification index of v_{i}/v and f_{i} is the residue degree of v_{i}/v ([2], VI, §8.3, Cor. 3). Since v_{i} extends v, the value group of v_{i} is \frac{1}{e_{i}} ℤ. For any x ∈ L, we have

(1)  v(N_{L/F} (x)) = ∑_{i=1}^{s} e_{i}f_{i}v_{i}(x),

by, [2], VI, §8.5, Cor. 2. Now by the Approximation Theorem ([2], VI, §7.2, Cor. 1) one can choose x ∈ L such that v_{1}(x) = 1/e_{1} and v_{i}(x) = 0 for all i > 1. Thus by (1), v(N_{L/F} (x)) = f_{1}. But since N_{L/F} (x) ∈ F^{∗d}, we must have d | v(N_{L/F} (x)) = f_{1} ≤ d. This, combined with ∑_{i=1}^{s} e_{i}f_{i} = d with all e_{i} ≥ 1 and f_{i} ≥ 1, forces f_{1} = d = [L:F] and s = 1, as desired. □

Using Proposition 4, we obtain the following Theorem which provides a further class of fields with the NKNT property which is not covered by the main Theorem 7. For example, it shows that if F is NKNT, then so is the Laurent power series field F ((x)).

Theorem 5. Let F be a discrete valued field with residue field \overline{F} such that char(F) = char(\overline{F}). If \overline{F} is NKNT, then so is F.

Proof. Suppose there is a noncommutative division algebra D finite dimensional over F with center K such that NK_{1}(D) = 1. We can assume K = F. Since by Lemma 1, NK_{1} respects the primary decomposition of D, it is enough to consider the case when ind(D) = p^{k}, where p is prime and k ≥ 1. If char(F) = p, then D is a p-algebra and by Corollary 3, NK_{1}(D) is nontrivial. Thus we may assume that char(F) ≠ p. Hence, every subfield of D containing F is separable over F.

Let d = p^{k} = ind(D). Since NK_{1}(D) = 1, we have N_{L/F} (L^{∗}) = F^{∗}^{d} for every maximal subfield L of D. Since L is separable over F, the integral closure in L of the discrete valuation ring V_{F} of v on F is a finitely generated V_{F} -module. Thus by Proposition 4, v extends uniquely to any maximal subfield of D, with no ramification. So, v extends uniquely to any subfield of D. By the theorem of Ershov-Wadsworth (see [13], Th. 2.1 or [12]), it follows that v extends to a valuation on D, which is denoted again by v. Furthermore D is not ramified over F, i.e. the value group Γ_{D} of D coincides with the value group Γ_{F} of F. Let \overline{D} and \overline{F} be the residue division algebra and the residue field of the valuations on D and F. Since char(\overline{F}) = char(F) does not divide ind(D), the Ostrowski theorem for valued division algebras, [9] Th. 3, yields [D:F] = [\overline{D}:\overline{F}] |Γ_{D} : Γ_{F} | = [\overline{D}:\overline{F}]. Note also that [Z(\overline{D}):\overline{F}] | [D:F]; hence, Z(\overline{D}) is separable over \overline{F}. Thus, the surjectivity of the fundamental homomorphism Γ_{D}/Γ_{F} → Gal(Z(\overline{D})/\overline{F}), together with the fact that Z(\overline{D}) is normal and separable over \overline{F} and Γ_{D}/Γ_{F} = 1 force that Z(\overline{D}) = \overline{F} (see [13], Prop. 2.5 or [7], Prop. 1.7). Hence, ind(\overline{D}) = ind(D).

Since NK_{1}(\overline{D}) ≠ 1 and ind(\overline{D}) = ind(D) = d, there is \overline{a} ∈ \overline{D}^{∗} with Nrd\overline{D}(\overline{a}) ∉ \overline{F}^{d}. Let a be any inverse image of \overline{a} in the valuation ring V_{D} of D, and let L be any maximal subfield of D containing a. Let V_{L} be the valuation ring of the restriction of v to L, and let \overline{L} be the residue field of V_{L}. Because V_{L} is the unique extension of V_{F} to L, V_{L} is the integral closure of V_{F} in L; hence, it is a finitely-generated V_{F} -module. Since [\overline{D}:\overline{F}] = [D:F], we must have [\overline{L}:\overline{F}] = [L:F], showing that \overline{L} is a maximal subfield of \overline{D}. If \overline{b}_{1}, …, \overline{b}_{d} are any \overline{F} -vector space base of \overline{L}, then any inverse images b_{1}, …, b_{d} of the \overline{b}_{i} in the valuation ring V_{L} form a base of V_{L} as a free V_{F} -module. (The b_{i} generate V_{L} over V_{F} by Nakayama’s Lemma, and they are V_{F} -independent because V_{F} is a valuation ring and the \overline{b}_{i} are \overline{F} -independent.) By computing the norm N_{L/F} (a) as the determinant of the F -linear map multiplication by a using the base b_{1}, …, b_{d}, we obtain N_{L/F} (a) ∈ V_{F} and

\overline{N_{L/F} (a)} = N_{\overline{L}/\overline{F}} (\overline{a}) in \overline{F} .

Because we have assumed NK_{1}(D) = 1, we have

N_{L/F} (a) = Nrd_{D}(a) ∈ F^{∗d} ∩ V_{F} = V_{F}^{d} .

Hence,

Nrd\overline{D}(\overline{a}) = N_{\overline{L}/\overline{F}} (\overline{a}) = \overline{N_{L/F} (a)} ∈ \overline{V_{F}^{d}} = \overline{F}^{d} ,

contradicting the choice of \overline{a}. So, NK_{1}(D) ≠ 1, contradicting the choice of D. Thus, F is NKNT. □

Proposition 6. Let F ⊆ F (t) ⊆ L be fields with t transcendental over F and [L:F (t)] < ∞. If L = F (t)(α) for some α, then there is a discrete valuation ring V of F (t) with F ⊆ V such that V has an extension to a DVR W of L such that \overline{W} = \overline{V}. (In fact, there are infinitely many such V.)

Proof. Let R = F [t]. We can assume that α is integral over R. Let f = x^{n}+c_{n−1}x^{n−1}+…+c_{0} be the minimal polynomial of α over F (t). The integrality of α over R (with R integrally closed) assures that f ∈ R[x]. Let

P_{f} = {π ∈ F [t] | π is irreducible and monic in F [t] and π|f(r) for some r ∈ R}.

We will show that |P_{f}| = ∞. Assume first that c_{0} = 1, and write f = xh(x)+1 with h ∈ R[x] and deg(h) = n−1. Suppose |P_{f}| = {π_{1}, …, π_{k}}. Let s = tπ_{1} … π_{k}. Since h has only finitely many roots in R, there is a natural number ℓ with h(s^{ℓ}) ≠ 0. Then f(s^{ℓ}) = s^{ℓ}h(s^{ℓ}) + 1 has positive degree in t, so is not a unit of R. If p is an irreducible monic factor of f(s^{ℓ}), then p ∈ P_{f}, but p ∤ s, so p ∉ {π_{1}, …, π_{k}}, a contradiction. Hence P_{f} cannot be finite if c_{0} = 1.

Now assume c_{0} ≠ 1. Let f(c_{0}x) = c_{0}g(x). So g ∈ R[x] with deg(g) = deg(f) ≥ 1, and g has constant term 1. By the previous case, |P_{g}| = ∞. But sincef(c_{0}r) = c_{0}g(r), we have P_{g} ⊆ P_{f}. So, |P_{f}| = ∞, as claimed.

Now, take any π ∈ P_{f}, and let V be the DVR R_{(π)} which is the localization of R at its prime ideal (π). Let M = πV, which is the maximal ideal of V; so \overline{V} = V/M. Assume first that the ring V [α] is integrally closed.

Since π|f(r) for some r ∈ R, the image \overline{f} of f in \overline{V} [x] has a root \overline{r} in \overline{V}. Note that fF (t)[x] ∩ V [x] = fV [x], by the division algorithm as f is monic in V [x]. Hence, V [α] ≅ V [x]/fV [x] and

(2)  V [α]/MV [α] ≅ V [x]/(f, M) ≅ \overline{V} [x]/(\overline{f}).

Because \overline{f}(\overline{r}) = 0, x − \overline{r} is an irreducible factor of \overline{f} in \overline{V} [x]. Let N be the maximal ideal of V [α] containing MV [α] corresponding to (x − \overline{r})/(f) in \overline{V} [x]/(\overline{f}) in the isomorphism given by (2). Let W be the localization V [α]_{N}. Then W is a DVR, as V [α] is the integral closure of V in L. Furthermore, W ∩ F (t) = V and \overline{W} ≅ V [α]/N ≅ \overline{V} [x]/(x − \overline{r}) ≅ \overline{V}. Thus, the desired W exists for V = R_{(π)} whenever π ∈ P_{f} and R_{(π)}[α] is integrally closed.

To complete the proof we show that the needed integral closure property of R_{(π)}[α] occurs for all but finitely many π ∈ P_{f}. Let T be the integral closure of R in L; so T is a finitely generated R-module ([2], V, §3.2, Th. 2). We have R[α] ⊆ T, and T and R[α] each have quotient field L. So, T/R[α] is a finitely generated torsion R-module; hence it has nonzero annihilator in R. Therefore, there is b ∈ R with b ≠ 0 and bT ⊆ R[α]. Hence, R[α][1/b] = T [1/b], which is integrally closed. For any monic irreducible π ∈ R, if π ∤ b then the DVR R_{(π)} is a localization of R[1/b]. Hence, R_{(π)}[α] is a localization of R[1/b][α], so R_{(π)}[α] is integrally closed. There are only finitely many monic irreducibles of R dividing b. For all other π in the infinite set P_{f}, we have R_{(π)}(α) is integrally closed. □

Remark. For the result of Prop. 6, it is not sufficient to assume that [L:F (t)] < ∞. For example, suppose char(F) = p ≠ 0 and [F^{1/p}:F] ≥ p^{2}. Take any field K with F ⊆ K ⊆ F^{1/p} and p^{2} ≤ [K:F] < ∞, and let L = K(t). Take any discrete valuation ring V of F (t) and any extension of V to a DVR W of L. Identify \overline{V} and K with their canonical images in \overline{W}. Since \overline{V} = F (β) for some β, the Theorem of the Primitive Element shows that \overline{V} ∩ K = F (γ), for some γ ∈ K. Since γ^{p} ∈ F, we have [F (γ):F] ≤ p < [K:F], so \overline{V} doesn’t contain all of K. Because K ⊆ \overline{W}, this shows that \overline{W} ≠ \overline{V}.

Theorem 7. Let F be a field which is finitely generated but not algebraic over some subfield F_{0}. Then, F is NKNT.

Proof. We need to show that for each finite degree extension field K of F and each noncommutative finite dimensional division algebra D with center K, we have NK_{1}(D) is nontrivial. As in the proof of Theorem 5, we can assume that K = F and that F is a finite degree extension of F_{0}(t), with t transcendental over F_{0}. Since NK_{1} respects the primary decomposition of D, by Corollary 3 it suffices to consider the case where ind(D) = p^{k}, where p is a prime number with p ≠ char(F_{0}).

Let L be any maximal subfield of D and let S be the separable closure of F_{0}(t) in L. Then, S = F_{0}(t)(α) for some α. By Proposition 6, applied to the field extension F_{0}(t) ⊆ S, there is a DVR V of F_{0}(t) (with F_{0} ⊆ V) which has an extension to a DVR W of S with \overline{V} = \overline{W}. Because L is purely inseparable over S, W has a unique extension to a DVR Y of L, and \overline{Y} is purely inseparable over \overline{W}. Let Z = Y ∩ F, which is a DVR of F. Since \overline{W} = \overline{V} ⊆ \overline{Z} ⊆ \overline{Y}, we have \overline{Y} is purely inseparable over \overline{Z}. If char(F_{0}) = 0, it follows that \overline{Y} = \overline{Z}; hence [\overline{Y} : \overline{Z}] = 1 ≠ p^{k} = [L : F]. If char(F_{0}) = q ≠ 0, then [\overline{Y} : \overline{Z}] = q^{ℓ} for some ℓ ≥ 0. Since q ≠ p by hypothesis, we again have [\overline{Y} : \overline{Z}] ≠ [L : F].

Let V_{F} (resp. V_{L}) be the integral closure of V in F (resp. L), and let Z_{L} be the integral closure of Z in L. Because the integral closure of F_{0}[t] in F (resp. in L) is a finitely generated F_{0}[t]-module, by [2], V, §3.2, Th. 2, and V is a localization of F_{0}[t] (or F_{0}[t^{−1}]), V_{F} and V_{L} are finitely generated V -modules, so V_{L} is a finitely generated V_{F} -module. Then, as Z is a localization of V_{F}, Z_{L} is a finitely generated Z-module. Since the conclusion of Proposition 4 fails for Z ⊆ Y in the field extension F ⊆ L, we have F^{∗}^{p^{k}} ⫋ N_{L/K}(L^{∗}) ⊆ Nrd(D^{∗}), showing that NK_{1}(D) is nontrivial. □

Corollary 8. If D is a noncommutative division algebra whose center is finitely generated over its prime field or over an algebraically closed field, then NK_{1}(D) ≠ 1. Hence, CK_{1}(D) ≠ 1 and D^{∗} contains a maximal proper normal subgroup.

Proof. This is immediate from the Theorem and the comments in the introduction. □

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Department of Pure Mathematics, Queen’s University, Belfast BT7 1NN, U.K.

E-mail address: r.hazrat@qub.ac.uk

Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112, U.S.A.

E-mail address: arwadsworth@ucsd.edu