Subfields and Splitting Fields of Division Algebras

Abstract

In this chapter we apply the machinery developed in previous chapters to analyze the subfields and splitting fields of division algebras over a Henselian field F. In §9.1 we give properties of the splitting fields of tame division algebra D with center F, with particularly strong criteria proved if D is inertial or totally ramified over F. This leads to explicit constructions of several interesting examples of division algebras, including noncyclic division algebras of degree p ² with no maximal subfield of the form $F(\!\sqrt[p^{2}]{a})$ in Examples 9.15, 9.17, and 9.18; noncyclic p-algebras in Ex. 9.26; noncrossed product algebras including universal division algebras in Th.  9.30 and division algebras over Laurent series over $\mathbb {Q}$, noncrossed products whose degree exceeds the exponent in Cor. 9.46; and crossed product division algebras with only one Galois group for all maximal subfields Galois over the center in Prop. 9.28[9.28].

Appendices

Exercises

Exercise 9.1

Prove that the degree-p ² noncrossed product division algebra  𝒮(A;K) of Prop. 9.37 has exponent p ².

Exercise 9.2

Let M/F be an elementary abelian Galois extension of degree p ², for some prime p, and let K⊆M be a subfield of codimension p. Suppose F contains a primitive p-th root of unity ω (hence $\operatorname {\mathit{char}}F\neq p$). We say that a cyclic F-algebra A of degree p ² is adapted to M through K if A≅(L/F,σ,a) for some cyclic Galois extension L/F and some a∈F ^× such that $L^{\sigma^{p}}\cong K$ and $K(\sqrt[\uproot 2 p]{a})\cong M$. Show that there exists a cyclic F-algebra adapted to M through K if and only if ω is a norm from K to F. When this condition holds, the Brauer classes of cyclic F-algebras adapted to M through K form a coset modulo Dec(M/F).

Exercise 9.3

Let F be a Henselian-valued field containing a primitive p-th root of unity for some prime p, with $p\neq \operatorname {\mathit{char}}\overline{F}$. Let D be a tame semiramified central division algebra over F. Assume $\operatorname {\mathit{deg}}D=p^{2}$ and $\operatorname {\mathit{exp}}(\Gamma_{D}/\Gamma_{F})=p$, so $\overline{D}/\overline{F}$ is an elementary abelian Galois extension of degree p ². Show that D is cyclic if and only if sp(D) is the coset of Brauer classes of cyclic $\overline{F}$-algebras adapted to $\overline{D}$ through some subfield of codimension p (see Exercise 9.2). When this condition holds, show that D is a cyclic algebra adapted to any inertial lift of $\overline{D}$.

Notes

§9.1: Proposition 9.2 gives information on the splitting fields of an inertially split central division algebra D over a Henselian field F. In the more specific case where the valuation on F is also discrete of rank 1 and $\operatorname {\mathit{ind}}E = \operatorname {\mathit{exp}}E$ for all central division algebras E over $\overline{F}$, Brussel has given in [40] a much more complete description of the splitting fields of D. This applies, e.g., when F=k((x)) for k a global or local field.

Corollary 9.5: It was proved in Tignol–Amitsur [244, Th.] (= [10, pp. 565–571]) that for a tame division algebra D over a strictly Henselian field F every splitting field of D finite-dimensional over F contains a maximal subfield of D. The criterion for splitting in terms of the canonical pairing on Γ _D was given by Tignol–Wadsworth [245, Prop. 4.5].

§9.2.1: The first example of a noncyclic division algebra with pure maximal subfield is a degree 4 algebra constructed by Albert [3]. This algebra has exponent 4. Another example, with degree 4 and exponent 2, was given by Dubisch  [66]. Examples of degree p ² for p≠2 remained elusive until the work of Matzri et al. [138]. Note that, by contrast, a division p-algebra with a pure maximal subfield is necessarily cyclic, by a theorem of Albert [4, Th. VII.26].

Lemma 9.11 is due to Albert [2, Th. 3] (see also [5, p. 468, Th. 3]). It can also be proved by computing the connecting map $H^{1}(\operatorname {\mathcal {G}}(k),\mathbb {Z}/p^{n}\mathbb {Z}) \to H^{2}(\operatorname {\mathcal {G}}(k),\mathbb {Z}/p\mathbb {Z})$ in the Galois cohomology exact sequence associated to the short exact sequence of trivial Galois modules

$$0\,\longrightarrow\,\mathbb {Z}/p\mathbb {Z}\,\longrightarrow\, \mathbb {Z}/p^{n+1}\mathbb {Z}\,\longrightarrow\,\mathbb {Z}/p^n\mathbb {Z}\,\longrightarrow\, 0. $$

Proposition 9.14 and Ex. 9.15 were given by Matzri–Rowen–Vishne [138]. They raised as an open question the existence of noncyclic division algebras of odd prime exponent with pure maximal subfields [138, Question 4.4]. Example 9.18 settles this question.

§9.2.2: The notion of weak coset is inspired by the degeneracy condition on the matrix (u _ij ) associated to a 2-cocycle of an abelian Galois group, as defined by Amitsur–Saltman [12] (= [10, pp. 441–452]) and revisited by McKinnie  [139, Def. 1.5]. (See the Notes of Ch. 8 for the definition of the matrix (u _ij ) and the correspondence with cosets modulo the Dec subgroup.) Proposition 9.20 is essentially due to Amitsur–Saltman [12, Lemma 3.1]; it has been rephrased by Boulagouaz–Mounirh [29, §3] and by McKinnie [139, Lemma 1.7].

§9.3: Th. 9.24 is due to Mounirh [165, Prop. 1.2]. Th. 9.27 is proved in McKinnie [140, Th. 1.2.1]. It was first established by Saltman [218, Th. 3.2] in the particular case where D is a “generic abelian crossed product,” i.e., obtained by the 𝒮′ construction of §8.4.5, under a somewhat stronger hypothesis on the specialization coset sp(D).

A finite group G is said to be ‘‘rigid” (for a field F) if there is a crossed product algebra over F with group G and no other group. Saltman showed in [218] that every finite noncyclic abelian p-group is rigid for some field of characteristic p (see Prop. 9.28). It is easy to see from Cor. 9.5 that every elementary abelian p-group is rigid for a valued field (F,v) with Γ _F sufficiently large and $\overline{F}$ containing a primitive p-th root of unity: take a division algebra  D that is a tensor product of symbol algebras of degree p and with a valuation extending v such that D is totally ramified over F. This was essentially proved by Amitsur in  [6] (= [10, pp. 419–431]). Brussel gave in [37], [39] examples of rigid nonabelian p-groups for k(x) and k((x)), where k is an algebraic number field containing no primitive p-th root of unity.

§9.4: Amitsur proved Th. 9.29 in his landmark noncrossed product paper [6] (= [10, pp. 419–431]), and used it to prove that $\mathit{UD}(\mathbb {Q},n)$ is not a crossed product if p ² | n for some odd prime p or if 8 | n. His examples of division algebras that are crossed products only with certain groups included some division algebras  D over the strictly Henselian field ${F = k((x_{1}\!))\!\!\mspace{1mu}\ldots\!\!\mspace{1mu}((x_{m}\!))}$, where k is algebraically closed. While he noted that such a D has a valuation extending the usual valuation on F, his information about Galois groups of maximal subfields of D was encoded in the diagonal entries of certain integer-valued matrices. In fact, those diagonal entries represent the invariant factors of Γ _D /Γ _F . While the valuation theory was not explicit in Amitsur’s paper, the further work on noncrossed products that it spawned was a major impetus in the development of noncommutative valuation theory. (The other major impetus was the work of Platonov and Yanchevskiĭ on SK ₁ of division algebras. See Ch. 11 below.)

After Amitsur’s paper, several authors obtained other examples of noncrossed product universal division algebras UD(F,n) by combining Amitsur’s Theorem 9.29 with further examples of crossed product division algebras with limited groups. This included Schacher–Small [224] (for $\operatorname {\mathit{char}}F \ne 0$ and certain n prime to $\operatorname {\mathit{char}}F$), Amitsur [7] (=  [10, pp. 433–439]), Risman  [206], Fein–Schacher [78], and Saltman [218] (for p-algebras).

The first examples of noncrossed product algebras not built from generic division algebras were given by Jacob–Wadsworth [105]. Once again the method was that of obtaining incompatible information about possible Galois groups of maximal subfields. But in this case the center F of the noncrossed product division algebra is an intersection F=K ₁∩K ₂, where each K _i is a field with Henselian valuation v _i , such that v ₁| _F and v ₂| _F are independent. For a prime p with $\overline {K_{1}}$ and $\overline {K_{2}}$ each containing a primitive p ⁿ-th root of unity, it was shown that there is a canonical isomorphism

$${}_{p^n}\!\operatorname {\mathit{Br}}(F) \, \cong \,{}_{p^n}\!\operatorname {\mathit{Br}}(K_1) \times \,{}_{p^n}\!\operatorname {\mathit{Br}}(K_2) $$

that is well-behaved with respect to index. When n≥3, this allows one to obtain central division algebras D over F such that each v _i | _F extends to D and the division algebras D⊗ _F K _i are tame and totally ramified over K _i , so crossed products, but only with different groups. Thus, D is not a crossed product.

The first examples of noncrossed products over ‘‘nice” and relatively ‘‘small” fields were given by Brussel [37]. In his examples, the ground field is the Laurent series field k((t)) or rational function field k(t) where k is a global field. The examples given in §9.4.1 and §9.4.2 illustrate his approach. The explicit examples given in these subsections are based on Hanke [92], [93], and Coyette [57].

Building on Brussel’s approach over k((t)), Hanke and Sonn [97] gave a very detailed analysis of which division algebras are noncrossed products over a field F with complete discrete rank 1 valuation with $\overline{F}$ a global field. For such a field there is the Witt decomposition (see (8.39))

$$\operatorname {\mathit{Br}}_ \mathit{tr}(F) \, \cong \, \operatorname {\mathit{Br}}(\overline{F}) \times \operatorname {\mathit{Hom}}^c(\operatorname {\mathcal {G}}(\overline{F}), \mathbb {Q}/\mathbb {Z}). $$

For any $\chi \in \operatorname {\mathit{Hom}}^{c}(\operatorname {\mathcal {G}}(\overline{F}), \mathbb {Q}/\mathbb {Z})$, they find a lower bound on the indexes of noncrossed products in $\operatorname {\mathit{Br}}(\overline{F})\times \{\chi\}$, and show that above that bound ‘‘nearly all” division algebras are noncrossed products. This work was generalized by Hanke–Neftin–Sonn [95] for inertially split division algebras over a Henselian field F with $\overline{F}$ a global field and Γ _F arbitrary; Hanke–Neftin–Wadsworth [96] generalized this still further for tame division algebras over a Henselian field with global residue field. Proposition 9.33 and Prop. 9.34 come from [96].

In later work on noncrossed products, Brussel [43] gave examples of such algebras over the rational function field F=k(t) where k is a local field such as $\mathbb {Q}_{p}$. For this he worked with the Gaussian extension to F, call it v, of the complete discrete valuation on k; so $\overline{F} = \overline{k}(t)$ where $\overline{k}$ is finite and $\Gamma_{F} = \mathbb {Z}$. Let (K,v _h ) be the Henselization (or the completion) of (F,v), so $\overline{K} = \overline{F} = \overline{k}(t)$, which is a global field. By lifting suitable families of cyclic field extensions and Brauer classes from $\overline{K}$ to K, he was able to build noncrossed product central division algebras E over K such that E has the form D⊗ _F K. Then, D is a noncrossed product division algebra over F. More recently, Brussel–McKinnie–Tengan [45] used the same general approach, combined with very substantial algebraic geometric machinery, to prove the existence of noncrossed product algebras over fields F of the following type: For any prime p, let X be a smooth curve over the p-adic ring $\mathbb {Z}_{p}$, and let F be the function field of X; so, F is an algebraic function field in one variable over the local field $\mathbb {Q}_{p}$. Let v be the discrete rank 1 valuation on F associated with the special fiber of X, and let $(\widehat{F}, \widehat{v})$ be the completion of (F,v). Then $v|_{\mathbb {Q}_{p}}$ is the complete discrete p-adic valuation on $\mathbb {Q}_{p}$; moreover, $\overline {\widehat{F}} = \overline{F}^{\,v}$, which is an algebraic function field in one variable over $\mathbb {F}_{p}$, so $\overline {\widehat{F}}$ is a global field. They showed that there is a highly noncanonical index-preserving map ${s\colon \operatorname {\mathit{Br}}'(\widehat{F}) \to \operatorname {\mathit{Br}}'(F)}$ splitting the map ${\operatorname {\mathit{ext}}_{\widehat{F}/F}\colon \operatorname {\mathit{Br}}'(F) \to \operatorname {\mathit{Br}}'(\widehat{F})}$, where $\operatorname {\mathit{Br}}'$ denotes the prime-to-p part of $\operatorname {\mathit{Br}}$. They used s to lift noncrossed products over $\widehat{F}$ to noncrossed products over F. Subsequently, Chen [51] proved analogous results by a somewhat different method in the more general situation that $\mathbb {Z}_{p}$ is replaced by an arbitrary complete discrete rank 1 valuation.

Amitsur generalized his Theorem 9.29 from maximal subfields of UD(F,n) to splitting fields. This was announced in [8] with a proof given in [9] (= [10, pp. 573–582]). Call a finite group H a splitting group of a central simple L-algebra A if there is a Galois field extension K of L such that K splits A and $\operatorname {\mathcal {G}}(K/L) \cong H$. Amitsur proved that for any infinite field F, if G is a splitting group of UD(F,n), then for every central simple algebra A of degree n over any field L containing F, there is a subgroup H _A of G that is a splitting group of A. Tignol–Amitsur [242], [243] (= [10, pp. 507–537, 539–563]) obtained lower bounds on |G| by analyzing subfields of Mal’cev–Neumann division algebras that are Kummer extensions of the center; they reduced the question of splitting groups to Galois groups of maximal subfields using Cor. 9.5. Tignol [237] thereafter fully characterized the cyclic and Kummer subfields of Mal’cev–Neumann algebras. His results were generalized by Morandi–Sethuraman [161], [162] for division algebras of the form S⊗ _F T over a Henselian field F, where S is inertially split and T is tame and totally ramified over F; these results were later generalized to arbitrary tame division algebras over a Henselian field by Mounirh [166] using graded methods.

In his work on noncrossed products of degree exceeding the exponent, Saltman proved the following general result [219, Lemma 3]: If a division algebra D with center F has a maximal subfield normal over F, then D has a maximal subfield Galois over F (so D is a crossed product). The nontrivial proof of this depends on a delicate analysis of presentations of cyclic p-algebras that is given in [217, Lemma 6] (or see Jacobson [108, Lemma 4.4.16]). Hanke noted in  [94, Prop. 2] that Saltman’s result holds in the more general setting replacing F=Z(D) with F a subfield of Z(D) over which Z(D) is separable of finite degree. Using this (and Exercise 5.6 above), one can eliminate the hypothesis on $\operatorname {\mathit{char}}\overline{F}$ in Prop. 9.32—the conclusion then becomes: $\overline{D}$ has maximal subfield Galois over $\overline{F}$. Proposition 9.32 comes from Jacob–Wadsworth [106, Th. 5.15(b)]. Again using Saltman’s result, it was shown by Hanke–Neftin–Wadsworth [96] that Prop. 9.34 holds with the ‘‘tamely ramified” condition in (a) deleted. This yields a full criterion for when a tame division algebra over a Henselian field is a crossed product, expressed entirely in terms of residue data. By another application of Saltman’s result, one can eliminate the hypothesis that $\operatorname {\mathit{char}}k\nmid \operatorname {\mathit{deg}}D$ in Lemma 9.43 and Prop. 9.44.

For further discussion of noncrossed products and related questions, see Auel et al. [17].