We consider Galois embedding problems G↠H≅Gal(X/Z)G\twoheadrightarrow H\cong \operatorname {Gal}(X/Z) such that a Galois embedding problem G↠Gal(Y/Z)G\twoheadrightarrow \operatorname {Gal}(Y/Z) is solvable, where Y/ZY/Z is a Galois subextension of X/ZX/Z. For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative kk-algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic pp-groups to which the reduction theorem applies.
An isomorphism is given between the trace bilinear form of Mestre’s An{A_n} extensions over Q(t)\mathbb {Q}(t) and a certain quadratic form over Q\mathbb {Q} with base field lifted to Q(t)\mathbb {Q}(t). This reduces the problem of constructing A~n{\tilde A_n} extensions from Mestre’s An{A_n}’s to that of diagonalizing certain forms over Q\mathbb {Q}. The result expands a result of Schneps.
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).
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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).
Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G F of fields F containing a primitive p th root of unity ξ p . In this paper we present several constraints on such G F , using restrictions on the cohomology of index p normal subgroups from N. Lemire, J. Mináč , and J. Swallow , Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics, J. reine angew. Math. 613 (2007), 147–173. In section 1 we classify all maximal p -elementary abelian-by-order p quotients of these G F . In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G F . Finally, in section 3 we construct a new family of pro- p -groups which are not absolute Galois groups over any field F .
Recently the Galois module structure of square power classes of a field $K$ has been computed under the action of $\text{Gal}(K/F)$ in the case where $\text{Gal}(K/F)$ 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 $\text{Br}(F)$, and show that all "unexceptional" summand types are possible.
