Automatic realizations of Galois groups with cyclic quotient of order {p^n}

Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p n

“…In particular, the case n = m = 1 was resolved in [13], the case m = 1 and n 1 (without the restriction ξ p ∈ E) in [11], and the case m 1 and n = 1 in [9]. As desired, these computed module structures have already led to some interesting results on the structure of absolute Galois groups: automatic realization results in [12,14], a generalization of Schreier's theorem in [6], a connection with Demuškin groups in [5], an interpretation of cohomological dimension in [8] and a characterization of certain groups that cannot appear as absolute Galois groups in [2].…”

Induced orthogonal representations of Galois groups

“…The results presented here can be established in two ways: by basic Kummer theory following [Wa], or by 'higher' Kummer theory developed in the works [MS1,MS2,MSS1,MSS2]. We will sketch most of the proofs, using the results of Mináč, Schultz and Swallow.…”

“…The author would like to thank an anonymous reviewer of an earlier version of this paper for suggesting the modified proofs in Section 3, based on the papers [MS1,MS2,MSS1,MSS2].…”

“…According to [MSS2], the non-abelian extensions L/F are in a bijective correspondence with the indecomposable dimension 2 submodules of J , and the Galois group G = Gal(L/F ) is isomorphic either to the modular group, or to the semidirect product C q (C p ) 2 . 2…”

Parameterizing solutions to any Galois embedding problem overZ/pnZwith elementary p -abelian kernel

“…Jensen has written a number of excellent articles on automatic realizations, including [13,14,15], and there are other automatic realizations considered in [3,9,19,22,34]. It is worth noting that prior to [30], the known automatic realization results for nonabelian p-groups with p > 2 were extremely limited, and seem to not involve groups of order larger than p 4 ; the module-theoretic machinery used in [30] and this paper, on the other hand, provide several infinite classes of automatic realizations for non-abelian p-groups, and even many cases where one (relatively) small p-group automatically realizes a (relatively) larger p-group.…”