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
Étale Algebras
The definition of étale algebras over a field and a characterization via diagonalizability
5
Separable Extensions
Characterization of separable extensions through étale algebras
6
Separable Degree
The decomposition of separable and inseparable degrees
7
Galois Extensions
Definition of Galois extensions satisfying normality and separability
8
Properties of Galois Groups
Structure of infinite Galois groups with the Krull topology