Topological Galois theory

Moore and Seiberg equations, Topological Field Theories, and Galois Theory

The following text is an outline of a review paper on Moore and Seiberg’s equations, Topological (Projective) Field Theories in three dimensions and their relationship with Grothendieck’s second paragraph of the “Esquisse d’un Programme”. Because of schedule and length problems this review paper has not been included in this volume. First of all, we recall the construction of projective topolog...

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...

Galois Theory