Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).
Deep Dive into Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p^n.
Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E’/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).
Absolute Galois groups capture a great deal of the arithmetic and algebraic properties of their underlying fields, though they are notoriously intractable to compute. For a given field E, one must often be satisfied with studying invariants attached to the corresponding absolute Galois group G E , and in this respect the Galois cohomology groups H i (G E , A) for various G E -modules A are frequent subjects of investigation. Of particular interest are the groups H m (G E , µ p ) for a fixed prime p, where µ p represents the group of pth roots of unity in G E .
When E is itself a Galois extension of a field F , the action of Gal(E/F ) on E × induces a natural action on H m (G E , µ p ). Combined with the F p -action on these cohomology groups, this naturally leads one to study these Galois cohomology groups as F p [Gal(E/F )]-modules. In particular, one expects that this Galois module structure will provide insight into the corresponding absolute Galois group G E . This program has been carried out in several cases where Gal(E/F ) ≃ Z/p n Z and E contains a primitive pth root of unity ξ p . In particular, the case n = m = 1 was resolved in [MS2], m = 1 and n ≥ 1 (without the restriction ξ p ∈ E) in [MSS1], and m ≥ 1 and n = 1 in [LMS2]. As desired, these computed module structures have already led to some interesting results on the structure of absolute Galois groups: automatic realization results in [MS3,MSS2], a generalization of Schreier’s Theorem in [LLMS2], a connection with Demuškin groups in [LLMS1], an interpretation of cohomological dimension in [LMS], and a characterization of certain groups which cannot appear as absolute Galois groups in [BLMS].
The goal of this paper is to begin the investigation of a unified understanding of the structures already computed by determining some important results in the case m ≥ 1 and n ≥ 1. We shall focus on the case p > 2 in this paper. In much the same way that this problem is the unification of the problems considered in [LMS2] and [MSS1], so too will the methodology in our solution be a combination of their individual strategies. Indeed, careful refinements of the arguments from [LMS2], together with the appropriate module-theoretic results, will already be enough to give us the following Theorem 1.1. Let p > 2 be a given prime. If Gal(E/F ) ≃ Z/p n Z and ξ p ∈ E, then the F p [Gal(E/F )]-module H m (G E , µ p ) is a direct sum of indecomposable summands which are either of dimension p n or of dimension at most 2p n-1 .
Since there are p n isomorphism classes of indecomposable F p [G]modules -one for each cyclic submodule of F p -dimension i, 1 ≤ i ≤ p n -this result shows that the decomposition of H m (G E , µ p ) is relatively sparse.
A more refined decomposition is available, however, if we impose an additional assumption on the extension E/F . When Gal(E/F ) ≃ Z/p n Z and ξ p ∈ E, we say that E/F is an embeddable extension if E/F is an intermediate extension in a larger Galois extension E ′ /F so that
where the horizontal arrows are the natural projections.
In the case of embeddable extensions, we can then use results from [MSS1] -particularly the properties of so-called “exceptional” elements of E (see Proposition 2.10) -to give the following result. In the statement of the result, we use E j to denote the intermediate field of degree p j over F within the extension E/F . Theorem 1.2. Let p > 2 be a given prime. If E/F is an embeddable extension and a n is an exceptional element, then as an F p [Gal(E/F )]module we have
where
Though the strategies for embeddable extensions cannot be translated directly into a decomposition of the Galois module structure of H m (G E , µ p ) when E/F is not embeddable, this is nonetheless an important step towards resolving the more general case. As an indication of this, we note that for a non-embeddable extension E/F , any proper subextension is embeddable. For “bottom-up” inductive arguments (i.e., those which rely on studying subextensions which share a common base field), then, the embeddable case is of critical importance. These kinds of arguments were already used to great effect in resolving the case m = 1, n > 1 in [MSS1], so it is likely that a resolution of the general (non-embeddable) case for higher cohomology will also include this strategy.
Section 2 outlines the basic ingredients necessary for the proofs of the main theorems, recalling important facts about Galois cohomology, module theory and field theory. Section 3 then gives a description of a submodule Γ(m, n) ⊆ H m-1 (G E n-1 , µ p ) which is critical for our inductive approach. Building on these results, Section 4 describes the major technical results needed to provide a proof of Theorem 1.2 in Section 5.
Remark 1.3. Though the proof of Theorem 1.2 relies on working in an embeddable extension, the other machinery we develop holds for extensions E/F with Gal(E/F ) ≃ Z/p n Z and ξ p ∈ E (p > 2 a prime) without insisting on embeddability.
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