Algebra: Fields and Galois Theory

Core idea

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

• Transcendental element
• Algebraic extension
• Transcendental extension
• Finitely generated field extension

Splitting Fields and Closures

• Splitting field
• Algebraic closure
• Normal extension

Separability and Normality

• Separable element
• Separable extension
• Inseparable extension
• Perfect field
• Galois extension

Galois Theory

• Galois group
• Fixed field
• Field embedding
• Field automorphism

Trace, Norm, Discriminant

• Trace (field extension)
• Norm (field extension)
• Discriminant (field extension)

Finite Fields

• Finite field
• Frobenius endomorphism
• Cyclotomic polynomial

Global and Local Fields

• Global field
• Number field
• Global function field
• Place of a global field
• Completion of a field at a place
• Local field
• Archimedean local field
• Nonarchimedean local field
• $p$-adic field

Cyclotomic Extensions

• Primitive root of unity
• Cyclotomic extension

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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)

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Lemmas

• Dedekind's independence lemma
• Separable polynomial has distinct roots
• Separability preserved under towers
• Normality = being a splitting field

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Propositions

• Finite extension over perfect field is separable
• Finite fields are perfect
• Separable + normal ⇔ Galois
• Trace/norm in towers
• Splitting field degree bounds

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Corollaries

• Uniqueness of splitting fields up to K-isomorphism
• Uniqueness of algebraic closures
• Galois correspondence (subgroups ↔ intermediate fields)
• $\lvert\operatorname{Gal}(L/K)\rvert=[L:K]$ for Galois extensions
• Finite field Galois groups are cyclic
• Existence and uniqueness of 𝔽_{p^n}
• Multiplicative group of finite field is cyclic

Imaginary quadratic arithmetic

The Bianchi collection develops the number-theory prerequisites and their geometric applications.

• Imaginary quadratic field
• Ring of integers
• Number-field discriminant
• Ideal class group
• Dedekind zeta function