ENGLISH

Galois Theory

By Ian Nicholas Stewart

Book information

Publisher
CRC Press
Year
2015
ISBN
9781482245837, 1482245833
Language
english
Format
PDF
Filesize
4 MB (4310610 bytes)
Edition
4
Pages
344\323
Time added
2020-07-26 18:59:23

Description

Since 1973, Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fourth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today's algebra students. New to the Fourth EditionThe replacement of the topological proof of the fundame Contents (vii) Acknowledgements (xi) Preface to the First Edition (xiii) Preface to the Second Edition (xv) Preface to the Third Edition (xvii) Preface to the Fourth Edition (xxi) Historical Introduction (1) Polynomial Equations (2) The Life of Galois (5) 1. Classical Algebra (17) 1.1 Complex Numbers (18) 1.2 Subfields and Subrings of the Complex Numbers (18) 1.3 Solving Equations (22) 1.4 Solution by Radicals (24) Exercises (31) 2. The Fundamental Theorem of Algebra (35) 2.1 Polynomials (35) 2.2 Fundamental Theorem of Algebra (39) 2.3 Implications (42) Exercises (44) 3. Factorisation of Polynomials (47) 3.1 The Euclidean Algorithm (47) 3.2 Irreducibility (51) 3.3 Gauss's Lemma (54) 3.4 Eisenstein's Criterion (55) 3.5 Reduction modulo 𝑝 (57) 3.6 Zeros of Polynomials (58) Exercises (60) 4. Field Extensions (63) 4.1 Field Extensions (63) 4.2 Rational Expressions (66) 4.3 Simple Extensions (67) Exercises (69) 5. Simple Extensions (71) 5.1 Algebraic and Transcendental Extensions (71) 5.2 The Minimal Polynomial (72) 5.3 Simple Algebraic Extensions (73) 5.4 Classifying Simple Extensions (75) Exercises (77) 6. The Degree of an Extension (79) 6.1 Definition of the Degree (79) 6.2 The Tower Law (80) Exercises (84) 7. Ruler-and-Compass Constructions (87) 7.1 Approximate Constructions and More General Instruments (89) 7.2 Constructions in ℂ (90) 7.3 Specific Constructions (94) 7.4 Impossibility Proofs (99) 7.5 Construction From a Given Set of Points (101) Exercises (102) 8. The Idea Behind Galois Theory (107) 8.1 A First Look at Galois Theory (108) 8.2 Galois Groups According to Galois (108) 8.3 How to Use the Galois Group (110) 8.4 The Abstract Setting (111) 8.5 Polynomials and Extensions (112) 8.6 The Galois Correspondences (114) 8.7 Diet Galois (116) 8.8 Natural Irrationalities (121) Exercises (125) 9. Normality and Separability (129) 9.1 Splitting Fields (129) 9.2 Normality (132) 9.3 Separability (133) Exercises (135) 10. Counting Principles (137) 10.1 Linear Independence of Monomorphisms (137) Exercises (142) 11. Field Automorphisms (145) 11.1 𝐾-Monomorphisms (145) 11.2 Normal Closures (146) Exercises (149) 12. The Galois Correspondence (151) 12.1 The Fundamental Theorem of Galois Theory (151) Exercises (153) 13. A Worked Example (155) Exercises (159) 14. Solubility and Simplicity (161) 14.1 Soluble Groups (161) 14.2 Simple Groups (164) 14.3 Cauchy's Theorem (166) Exercises (168) 15. Solution by Radicals (171) 15.1 Radical Extensions (171) 15.2 An Insoluble Quintic (176) 15.3 Other Methods (178) Exercises (179) 16. Abstract Rings and Fields (181) 16.1 Rings and Fields (181) 16.2 General Properties of Rings and Fields (184) 16.3 Polynomials Over General Rings (186) 16.4 The Characteristic of a Field (187) 16.5 Integral Domains (188) Exercises (191) 17. Abstract Field Extensions (193) 17.1 Minimal Polynomials (193) 17.2 Simple Algebraic Extensions (194) 17.3 Splitting Fields (195) 17.4 Normality (197) 17.5 Separability (197) 17.6 Galois Theory for Abstract Fields (202) Exercises (203) 18. The General Polynomial Equation (205) 18.1 Transcendence Degree (205) 18.2 Elementary Symmetric Polynomials (208) 18.3 The General Polynomial (209) 18.4 Cyclic Extensions (211) 18.5 Solving Equations of Degree Four or Less (214) Exercises (218) 19. Finite Fields (221) 19.1 Structure of Finite Fields (221) 19.2 The Multiplicative Group (222) 19.3 Application to Solitaire (224) Exercises (225) 20. Regular Polygons (227) 20.1 What Euclid Knew (227) 20.2 Which Constructions are Possible? (230) 20.3 Regular Polygons (231) 20.4 Fermat Numbers (235) 20.5 How to Draw a Regular 17-gon (235) Exercises (240) 21. Circle Division (243) 21.1 Genuine Radicals (244) 21.2 Fifth Roots Revisited (246) 21.3 Vandermonde Revisited (249) 21.4 The General Case (250) 21.5 Cyclotomic Polynomials (253) 21.6 Galois Group of ℚ(ξ): ℚ (255) 21.7 The Technical Lemma (256) 21.8 More on Cyclotomic Polynomials (257) 21.9 Constructions Using a Trisector (259) Exercises (263) 22. Calculating Galois Groups (267) 22.1 Transitive Subgroups (267) 22.2 Bare Hands on the Cubic (268) 22.3 The Discriminant (271) 22.4 General Algorithm for the Galois Group (272) Exercises (274) 23. Algebraically Closed Fields (277) 23.1 Ordered Fields and Their Extensions (277) 23.2 Sylow's Theorem (279) 23.3 The Algebraic Proof (281) Exercises (282) 24. Transcendental Numbers (285) 24.1 Irrationality (286) 24.2 Transcendence of e (288) 24.3 Transcendence of π (289) Exercises (292) 25. What Did Galois Do or Know? (295) 25.1 List of the Relevant Material (296) 25.2 The First Memoir (296) 25.3 What Galois Proved (297) 25.4 What is Galois Up To? 299) 25.5 Alternating Groups, Especially 𝔸₅ (301) 25.6 Simple Groups Known to Galois (302) 25.7 Speculations about Proofs (303) Exercises (307) References (309) Galois Theory (309) Additional Mathematical Material (310) Historical Material (312) The Internet (313) iPad App (314)

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