Localic Galois theory

Localic Galois Theory

In this article we prove the following: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point. We explain and give the necessary definitions to understand this statement. The hard direction in this equivalence was first proved in print in [4], Theorem 1, Section 3, Chapte...

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