The power classes of a field are well-known for their ability to parameterize elementary $p$-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple. Suppose $m,n \in \mathbb{N}$ and $p$ is prime. In a recent paper, a surprising and elegant decomposition for $p^m$th power classes has been computed when the underlying Galois group is a cyclic group of order $p^n$. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when $p=2$, the structure of this exceptional summand was determined by a vector $\mathbf{a}\in \{-\infty,0,\dots,n\}^m$ and a natural number $d$. In this paper we give field-theoretic interpretations for $\mathbf{a}$ and $d$, showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to $K/F$.

Recently the Galois module structure of square power classes of a field K has been computed under the action of Gal ( K/F ) in the case where Gal( K/F ) is the Klein 4-group. Despite the fact that the modular representation theory over this group ring includes an infinite number of non-isomorphic indecomposable types, the decomposition for square power classes includes at most 9 distinct summand types. In this paper we determine the multiplicity of each summand type in terms of a particular subspace of Br( F ), and show that all “unexceptional” summand types are possible.

Abstract For a Galois extension $K/F$ with $\text {char}(K)\neq 2$ and $\mathrm {Gal}(K/F) \simeq \mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}$ , we determine the $\mathbb {F}_{2}[\mathrm {Gal}(K/F)]$ -module structure of $K^{\times }/K^{\times 2}$ . Although there are an infinite number of (pairwise nonisomorphic) indecomposable $\mathbb {F}_{2}[\mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}]$ -modules, our decomposition includes at most nine indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types.

A powerful new perspective in the analysis of absolute Galois groups has recently emerged from the study of Galois modules related to classical parameterizing spaces of certain Galois extensions. The recurring trend in these decompositions is their stunning simplicity: almost all summands are free over some quotient ring. The non-free summands which appear are exceptional not only because they are different in form, but because they play the key role in controlling arithmetic conditions that allow the remaining summands to be easily described. In this way, these exceptional summands are the lynchpin for a bevy of new properties of absolute Galois groups that have been gleaned from these surprising decompositions. In one such recent decomposition, a remarkable new exceptional summand was discovered which exhibited interesting properties that have not been seen before. The exceptional summand is drawn from a particular finite family that has not yet been investigated. The main goal of this paper is to introduce this family of modules and verify their indecomposability. We believe this module will be of interest to people working in Galois theory, representation theory, combinatorics, and general algebra. The analysis of these modules includes some interesting new tools, including analogs of p-adic expansions.

Let p be prime, and $$n,m \in \mathbb {N}$$ . When K/F is a cyclic extension of degree $$p^n$$ , we determine the $$\mathbb {Z}/p^m\mathbb {Z}[\text {Gal}(K/F)]$$ -module structure of $$K^\times /K^{\times p^m}$$ . With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $$\text {Gal}(K/F)$$ . For fixed values of m and n, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when $$m,n>1$$ the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when $m,n>1$ the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.

In this paper, we use the Merkurjev-Suslin theorem to determine the structure of arithmetically significant Galois modules that arise from Kummer theory. Let K be a field of characteristic different from a prime l, n be a positive integer, and suppose that K contains the (l(n))th roots of unity. Let L be the maximal l(n)-elementary abelian extension of K, and set G=Gal(L vertical bar K). We consider the G-module J:L-x/l(n) and denote its socle series by J(m). We provide a precise condition, in terms of a map to H-3(G, Z/l(n)), determining which submodules of J(m-1) embed in cyclic modules generated by elements of J(m); therefore, this map provides an explicit description of J(m) and J(m)/J(m-1). The description of J(m)/J(m-1) is a new non-trivial variant of the classical Hilbert's Theorem 90. The main theorem generalizes a theorem of Adem, Gao, Karaguezian and Minac that deals with the case m=l(n)=2, and also ties in with current trends in minimalistic birational anabelian geometry over essentially arbitrary fields.

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).

We determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.

We establish automatic realizations of Galois groups among groups M\rtimes G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].

Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several generalizations, discerning connections among multiplicative groups of fields, values of binary quadratic forms, a bit of module theory over group rings, and even Galois cohomology.

Let p be a prime and F a field containing a primitive pth root of unity. Then for n in N, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is <=n if and only if the corestriction maps H^n(H,Fp) -> H^n(G,Fp) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_Fp H^n(H,Fp) of Schreier's formula for dim_Fp H^1(H,Fp).

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G_F of F. Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H^n(U,Fp) as an Fp[G_F/U]-module for all n in N. Previously this structure was known only for n=1, and until recently the structure even of H^1(U,Fp) was determined only for F a local field, a case settled by Borevic and Faddeev in the 1960s. We apply these results to study partial Euler-Poincare characteristics of open subgroups N of the maximal pro-p quotient T of G_F. We extend the notion of a partial Euler-Poincare characteristic to this case and we show that the nth partial Euler-Poincare characteristic Theta_n(N) is determined only by Theta_n(T) and the conorm in H^n(T,Fp).

Nous etablissons des realisations automatiques de groupes de Galois parmi les groupes M x G ou G est un groupe cyclique d'ordre p n , p premier, et M un groupe quotient de l'anneau Fp[G].

Let E be a cyclic extension of degree p^n of a field F of characteristic p. Using arithmetic invariants of E/F we determine k_mE, the Milnor K-groups K_mE modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, we show that each indecomposable summand of k_mE has Fp-dimension a power of p. That all powers p^i, i=0,1,...,n, occur for suitable examples is shown in a subsequent paper [MSS2], where additionally the main result of this paper becomes an essential induction step in the determination of K_mE/p^sK_mE as (Z/p^sZ)[Gal(E/F)]-modules for all m, s in N.

In the mid‐1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of p th‐power classes of cyclic extensions K/F of p th‐power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F . 2000 Mathematics Subject Classification 12F10 (primary), 16D70 (secondary)

Nous etablissons des realisations automatiques de groupes de Galois parmi les groupes M x G ou G est un groupe cyclique d'ordre p n , p premier, et M un groupe quotient de l'anneau Fp[G].

Let p be a prime and F(p) the maximal p-extension of a field F containing a primitive pth root of unity. We give a new characterization of Demuškin groups among Galois groups Gal(F(p)/F) when p=2, and, assuming the Elementary Type Conjecture, when p>2 as well. This characterization is in terms of the structure, as Galois modules, of the Galois cohomology of index p subgroups of Gal(F(p)/F).

. In the case p>2, each such quotient contains aunique closed index pelementary abelian subgroup. This seems to bethe first case in which one can completely classify nontrivial quotientsof absolute Galois groups by characteristic subgroups of normal sub-groups. In section 3 we derive analogues of theorems of Artin-Schreierand Becker for order pelements of certain small quotients of G