On inseparable Galois theory

Hopf Galois structures on primitive purely inseparable extensions

Let L/K be a primitive purely inseparable extension of fields of characteristic p, [L : K] > p, p odd. It is well known that L/K is Hopf Galois for some Hopf algebra H, and it is suspected that L/K is Hopf Galois for numerous choices of H. We construct a family of K-Hopf algebras H for which L is an H-Galois object. For some choices of K we will exhibit an infinite number of such H. We provide ...

On Quantum Galois Theory

The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models.” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2-dimensional CFT in the guise of vertex operator algebras (VOA)...

Notes on Galois Theory

Notes on Galois Theory

Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by AutE. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of AutE consisting of all automorphisms σ : E → E such that σ(a) = a for all a ∈ F . We write this last c...

Notes on Galois Theory

§1. Algebraic Extensions 2 §1.1. Field extensions 2 §1.2. Multiplicativity of degree 4 §1.3. Algebraic extensions 4 §1.4. Adjoining roots 5 §1.5. Splitting fields 5 §1.6. Algebraic closure 7 §1.7. Finite fields 9 §1.8. Composite field 9 §1.9. Exercises 10 §2. Galois Theory 11 §2.1. Separable extensions 11 §2.2. Normal extensions 13 §2.3. Main Theorem of Galois Theory 13 §2.4. Fields of invarian...