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Course 311: Hilary Term 2002 Part III: Introduction to Galois Theory D. R. Wilkins Contents 3 Introduction to Galois Theory 3.1 Rings and Fields . . . . . . . . . . . . . . . . . . . . 3.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quotient Rings and Homomorphisms . . . . . . . . . 3.4 The Characteristic of a Ring . . . . . . . . . . . . . . 3.5 Polynomial Rings . . . . . . . . . . . . . . . . . . . . 3.6 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . 3.7 Eisenstein’s Irreducibility Criterion . . . . . . . . . . 3.8 Field Extensions and the Tower Law . . . . . . . . . 3.9 Algebraic Field Extensions . . . . . . . . . . . . . . . 3.10 Ruler and Compass Constructions . . . . . . . . . . . 3.11 Splitting Fields . . . . . . . . . . . . . . . . . . . . . 3.12 Normal Extensions . . . . . . . . . . . . . . . . . . . 3.13 Separability . . . . . . . . . . . . . . . . . . . . . . . 3.14 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . 3.15 The Primitive Element Theorem . . . . . . . . . . . . 3.16 The Galois Group of a Field Extension . . . . . . . . 3.17 The Galois correspondence . . . . . . . . . . . . . . . 3.18 Quadratic Polynomials . . . . . . . . . . . . . . . . . 3.19 Cubic Polynomials . . . . . . . . . . . . . . . . . . . 3.20 Quartic Polynomials . . . . . . . . . . . . . . . . . . 3.21 The Galois group of the polynomial x4 − 2 . . . . . . 3.22 The Galois group of a polynomial . . . . . . . . . . . 3.23 Solvable polynomials and their Galois groups . . . . . 3.24 A quintic polynomial that is not solvable by radicals 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 4 5 7 7 10 12 12 14 16 21 24 25 27