Fields and galois theory

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• FIELDS AND GALOIS THEORY J.S. MILNE
• Contents
• 1. Extensions of Fields
• 1.1. Definitions. A field is a set F with two composition laws + and ・ such that
• 1.2. The characteristic of a field. The map
• 1.3. The polynomial ring F[X].
• 1.4. Factoring polynomials.
• 1.5. Extension fields; degrees.
• 1.6. Construction of some extensions.
• 1.7. Generators of extension fields.
• 1.8. Algebraic and transcendental elements.
• 1.9. Transcendental numbers.
• 1.10. Constructions with straight-edge and compass.
• 2. Splitting Fields; Algebraic Closures
• 2.1. Maps from simple extensions.
• 2.2. Splitting fields.
• 2.3. Algebraic closures.
• 3. The Fundamental Theorem of Galois Theory
• 3.1. Multiple roots.
• 3.2. Groups of automorphisms of fields.
• 3.3. Separable, normal, and Galois extensions.
• 3.4. The fundamental theorem of Galois theory.
• 3.5. Constructible numbers revisited.
• 3.6. Galois group of a polynomial.
• 3.7. Solvability of equations.
• 4. Computing Galois Groups.
• 4.1. When is Gf ⊂ An?
• 4.2. When is Gf transitive?
• 4.3. Polynomials of degree ≤ 3.
• 4.4. Quartic polynomials.
• 4.5. Examples of polynomials with Sp as Galois group over Q.
• 4.6. Finite fields.
• 4.7. Computing Galois groups over Q.
• 5. Applications of Galois Theory
• 5.1. Primitive element theorem.
• 5.2. Fundamental Theorem of Algebra.
• 5.3. Cyclotomic extensions.
• 5.4. Independence of characters.
• 5.5. Hilbert’s Theorem 90.
• 5.6. Cyclic extensions.
• 5.7. Proof of Galois’s solvability theorem.
• 5.8. The general polynomial of degree n.
• 5.9. Norms and traces.
• 5.10. Infinite Galois extensions (sketch).

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