Nonrigid constructions in Galois theory

Nonrigid Constructions in Galois Theory

The context for this paper is the Inverse Galois Problem. First we give an if and only if condition that a finite group is the group of a Galois regular extension of R(X) with only real branch points. It is that the group is generated by elements of order 2 (Theorem 1.1 (a)). We use previous work on the action of the complex conjugation on covers of P [FrD]. We also use Fried and Völklein [FrV]...

Galois Groups over Nonrigid Fields

Let F be a field with charF 6= 2. We show that there are two groups of order 32, respectively 64, such that a field F with char F 6= 2 is nonrigid if and only if at least one of the two groups is realizable as a Galois group over F . The realizability of those groups turns out to be equivalent to the realizability of certain quotients (of order 16, respectively 32). Using known results on conne...

Galois Theory

Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

Galois Theory

Proposition 1.3. Let φ be an automorphism of a field extension K/F , and f(x) ∈ F [x]. Let α1, . . . , αn be the roots of f(x) lying in K. Then φ permutes the set {α1, . . . , αn}. If also the set of αi generate K over F , then two automorphisms φ1, φ2 of K/F which agree on all the αi are equal. Thus, in this case we have an inclusion of Aut(K/F ) as a subgroup of Sym({α1, . . . , αn}) ∼= Sn. P...