1 Fields Definition of a field, prime field, and characteristic 2 Algebraic Extensions The definition and degree of algebraic extensions of fields 3 Algebraic Closure Algebraically closed fields and the existence of algebraic closures 4 Purely Inseparable Extensions Definition and role of p-radical extensions in Galois theory 5 Étale Algebras Definition of étale algebras over a field and characterization via diagonalizability 6 Separable Extensions Characterization of separable extensions through étale algebras 7 Separable Degree Decomposition of separable and inseparable degrees 8 Galois Extensions Definition of Galois extensions satisfying both normality and separability 9 Properties of Galois Groups Structure of infinite Galois groups with the Krull topology 10 Fundamental Theorem of Galois Theory Galois correspondence between subgroups and intermediate fields 11 Solvability by Radicals Solvability by radicals and solvable Galois group 12 Cyclotomic Extensions Irreducibility of cyclotomic polynomials and Galois groups of cyclotomic fields