1
Fields
Definition of a field, prime field, and characteristic
2
Algebraic Extensions
The definition and degree of algebraic extensions of fields
3
Algebraic Closures
The existence of algebraically closed fields and algebraic closures
4
Purely Inseparable Extensions
Definition and role of p-radical extensions in Galois theory
5
Étale Algebras
The definition of étale algebras over a field and a characterization via diagonalizability
6
Separable Extensions
Characterization of separable extensions through étale algebras
7
Separable Degree
The decomposition of separable and inseparable degrees
8
Galois Extensions
Definition of Galois extensions satisfying normality and separability
9
Properties of Galois Groups
Structure of infinite Galois groups with the Krull topology
10
The Fundamental Theorem of Galois Theory
The Galois correspondence between subgroups and intermediate fields