Course 311: Hilary Term 2006 Part IV: Introduction to Galois Theory
نویسندگان
چکیده
4 Introduction to Galois Theory 2 4.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Eisenstein’s Irreducibility Criterion . . . . . . . . . . . . . . . 6 4.4 Field Extensions and the Tower Law . . . . . . . . . . . . . . 6 4.5 Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . 8 4.6 Algebraically Closed Fields . . . . . . . . . . . . . . . . . . . . 11 4.7 Ruler and Compass Constructions . . . . . . . . . . . . . . . . 11 4.8 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.9 Normal Extensions . . . . . . . . . . . . . . . . . . . . . . . . 19 4.10 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.11 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.12 The Primitive Element Theorem . . . . . . . . . . . . . . . . . 25 4.13 The Galois Group of a Field Extension . . . . . . . . . . . . . 25 4.14 The Galois correspondence . . . . . . . . . . . . . . . . . . . . 28 4.15 Quadratic Polynomials . . . . . . . . . . . . . . . . . . . . . . 30 4.16 Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 30 4.17 Quartic Polynomials . . . . . . . . . . . . . . . . . . . . . . . 32 4.18 The Galois group of the polynomial x − 2 . . . . . . . . . . . 34 4.19 The Galois group of a polynomial . . . . . . . . . . . . . . . . 35 4.20 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.21 Solvable polynomials and their Galois groups . . . . . . . . . . 39 4.22 A quintic polynomial that is not solvable by radicals . . . . . 43
منابع مشابه
Course 311: Hilary Term 2006 Part VI: Introduction to Affine Schemes
6 Introduction to Affine Schemes 2 6.1 Rings and Modules of Fractions . . . . . . . . . . . . . . . . . 2 6.2 The Spectrum of a Unital Commutative Ring . . . . . . . . . 5 6.3 The Spectrum of a Quotient Ring . . . . . . . . . . . . . . . . 7 6.4 The Spectrum of a Ring of Fractions . . . . . . . . . . . . . . 9 6.5 Intersections of Prime Ideals . . . . . . . . . . . . . . . . . . . 11 6.6 Topolo...
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