On 2-groups as Galois groups. (English) Zbl 0849.12006

Ledet, Arne

Can. J. Math. 47, No. 6, 1253-1273 (1995).

Let \(L/K\) be a Galois extension and let \(1 \to A \to G \to G (L/K) \to 1\) be a nontrivial group extension. Then \(N/K\) is called a solution of the embedding problem defined above if \(N/K\) is a Galois extension with Galois group \(G(N/K) \simeq G\) and \(L \subseteq N\). This paper handles Galois extensions \(L/K\) such that \(G(L/K)\) is a direct product of groups, \(K\) has characteristic \(\neq 2\) and \(A\) has order 2. Then the obstruction to the considered embedding problem is a product of the obstructions to related embedding problems over the corresponding subextensions of \(L/K\) and some quaternion algebras. The author applies his result to a number of 2-groups, in particular to groups of order \(\leq 16\). He further studies automatic realisations of 2-groups as Galois groups. Some of these examples are well known, others are obtained independently [see H. Grundman, T. L. Smith and J. Swallow, Expos. Math. 13, 289-319 (1995; Zbl 0838.12004)].

Reviewer: M.Epkenhans (Paderborn)

MSC:

12F12	Inverse Galois theory
12F10	Separable extensions, Galois theory

Keywords:

Galois extensions; embedding problems; quaternion algebras; 2-groups as Galois groups

Citations:

Zbl 0838.12004