On $2$-groups as Galois groups

TL;DR: In this paper, a constructive approach to the inverse Galois problem is described, where given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G, and if there is such an extension, find an explicit polynomial over K whose group is the prescribed group G. The existence of such generic polyno-mials is discussed, and where they do exist, a detailed treatment of their construction is given.

TL;DR: In this paper, the realizability of a 2-group G over a field of characteristic 2 depends only on the minimal number of generators of G over the field of interest.

TL;DR: The genus 0 problem in 0 characteristic was studied in this paper, where the moduli space of exceptions to a specific diophantine outcome was investigated, e.g., if f is indecomposable and h is not a composition with f, then f(x) − h(y) is irreducible.

TL;DR: In this article, the authors use the Galois module structure for the classical parameterizing spaces for elementary p-abelian extensions of a field K to give necessary and sufficient conditions for the solvability of any embedding problem which is an extension of Z / p n Z with elementary pabelian kernel.

TL;DR: In this paper, the existence of a finite normal extension M of k, such that M contains L, and such that [M: L] = p, denoting by the Galois group of M/k, the extension is given by the class of c.

TL;DR: In this paper, the realizability of a 2-group G over a field of characteristic 2 depends only on the minimal number of generators of G over the field of interest.

TL;DR: In this article, conditions for the appearance or nonappearance of two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2 were examined.