Galois embedding problems with cyclic quotient of orderp

Abstract

Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F ) such that corresponding normal subgroups are indecomposable F p [Gal(K/F )]-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.

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

“…We take this approach in proving Theorem 1.1, which establishes automatic realizations for a useful family of finite metacyclic p-groups. Our methods extend those of [10], [11] and [12]. It is interesting to observe that, although not visible here, the essential fact underpinning our results is Hilbert 90.…”

The Sylow subgroups of the absolute Galois group Gal(Q)

“…We argue by induction on N with N = 0 being trivial. By induction D = P N ⊕ Q N , where P N is as in (14), and Q N has no summands of dimension ≤ N . The induction hypothesis applied to Q N also shows that U n−1 (Q N ) = 0 for all n ≤ N , as otherwise Q N would have direct Z-summands of dimension 1 ≤ d ≤ N .…”

“…The proof splits into finding the multiplicity of the infinite and finite indecomposable direct Z-summands. In contrast to the work of Mináč, Schultz and Swallow [14,13], this approach also allows dealing with modules over an infinite group such as Z, cf. Section 5.1…”

A characterization of Eisenstein polynomials generating extensions of degree p^2 and cyclic of degree p^3 over an unramified \mathfrak{p}-adic field

“…On the other hand Gal(K /K) has exponent p 2 so if K ⊂ LK m then also Gal(LK m /K) does, having Gal(K m /K) exponent p we would have a contradiction if Gal(L/K) had exponent p too. If Gal(L/F ) has greatest possible length m = p, then there is only one possibility for the isomorphism class, which is the wreath product of two cyclic groups of order p, see [MS05,Wat94].…”

“…Gal(L/F ) is a cyclic and indecomposable Gal(F/K)-module of length ≤ p, and the isomorphism class of the group Gal(L/F ) is identified by the length of Gal(L/F ) as Gal(F/K)-module, and by its exponent (see [MS05,Wat94]). …”