This section covers field theory and Galois theory: field extensions, algebraic and transcendental elements, splitting fields, algebraic closures, separability, normality, Galois extensions, and the fundamental theorem of Galois theory.
Definitions
Field Extensions
- Field extension
- Intermediate field
- Degree of a field extension
- Simple field extension
- Tower of fields
- Algebraic element
Algebraic and Transcendental Elements
Splitting Fields and Closures
Separability and Normality
Galois Theory
Trace, Norm, Discriminant
Finite Fields
Cyclotomic Extensions
Theorems
- Tower law (degree formula)
- Existence and uniqueness of splitting fields
- Existence of algebraic closures
- Primitive element theorem
- Fundamental theorem of symmetric polynomials
- Fundamental theorem of Galois theory
- Artin's theorem on fixed fields
- Existence and uniqueness of finite fields
- Cyclicity of multiplicative group of finite field
- Galois group of finite field is cyclic (Frobenius)
Lemmas
- Dedekind's independence lemma
- Separable polynomial has distinct roots
- Separability preserved under towers
- Normality = being a splitting field
Propositions
- Finite extension over perfect field is separable
- Finite fields are perfect
- Separable + normal โ Galois
- Trace/norm in towers
- Splitting field degree bounds
Corollaries
- Uniqueness of splitting fields up to K-isomorphism
- Uniqueness of algebraic closures
- Galois correspondence (subgroups โ intermediate fields)
- |Gal(L/K)| = [L:K] for Galois extensions
- Finite field Galois groups are cyclic
- Existence and uniqueness of ๐ฝ_{p^n}
- Multiplicative group of finite field is cyclic