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Galois Theory

Year 3 · 5 sections · 48 concepts · 81 practice questions

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What's covered

free1 · Field extensions11 concepts

Overview 1 · The plan of this section
Definition 1.1 · Algebraic and finite extensions
Proposition 1.2 · Finite extensions are algebraic
Proposition 1.3 · Adjoining finitely many algebraic elements
Theorem 1.4 · Extending embeddings to simple extensions
Theorem 1.5 · Existence of splitting fields
Theorem 1.6 · Uniqueness of splitting fields
Example 1.7 · Degrees of splitting fields
Definition 1.8 · Normal extensions
Theorem 1.9 · Normal extensions are splitting fields
Check 1.10 · Check: computing degrees and splitting fields

2 · Separability and automorphisms10 concepts

Overview 2 · The plan of this section
Definition 2.1 · Separable polynomials and extensions
Proposition 2.2 · The derivative test
Corollary 2.3 · Separability in characteristic zero and over finite fields
Definition 2.4 · The Galois group
Definition 2.5 · Fixed fields
Theorem 2.6 · Counting automorphisms of a splitting field
Theorem 2.7 · The primitive element theorem
Theorem 2.8 · Artin's theorem
Check 2.9 · Check: finding Galois groups of small extensions

3 · The Galois correspondence9 concepts

Overview 3 · The plan of this section
Definition 3.1 · Galois extensions
Theorem 3.2 · Characterisations of Galois extensions
Theorem 3.3 · The fundamental theorem of Galois theory
Proposition 3.4 · Normal subgroups and normal subextensions
Example 3.5 · The field Q(√2, √3)
Example 3.6 · The splitting field of x³ − 2
Theorem 3.7 · Galois groups of cyclotomic fields
Check 3.8 · Check: drawing the Galois correspondence

4 · Finite fields revisited6 concepts

Overview 4 · The plan of this section
Theorem 4.1 · Existence and uniqueness of finite fields
Theorem 4.2 · The Galois group of a finite field
Corollary 4.3 · Subfields of finite fields
Proposition 4.4 · Irreducible polynomials over F_p
Check 4.5 · Check: computations in finite fields

5 · Solvability12 concepts

Overview 5 · The plan of this section
Definition 5.1 · Solvable groups
Proposition 5.2 · Subgroups and quotients of solvable groups
Theorem 5.3 · S_n is not solvable for n ≥ 5
Definition 5.4 · Radical extensions and solvability by radicals
Proposition 5.5 · Galois groups of root extensions are abelian
Theorem 5.6 · Galois's theorem on solvability by radicals
Example 5.7 · A quintic not solvable by radicals
Definition 5.8 · Constructible numbers
Theorem 5.9 · Constructible numbers have degree a power of 2
Corollary 5.10 · Doubling the cube and trisecting the angle
Check 5.11 · Check: solvability and constructibility arguments

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Proposition 1.2 · Finite extensions are algebraic

Finite ⇒ algebraic, with $\deg m_\alpha\mid[K : F]$ .

Proof

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