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Galois Theory 5th edition [Minkštas viršelis]

4.22/5 (81 ratings by Goodreads)
(University of Warwick, UK)
  • Formatas: Paperback / softback, 386 pages, aukštis x plotis: 234x156 mm, weight: 600 g, 38 Line drawings, black and white; 38 Illustrations, black and white
  • Išleidimo metai: 07-Sep-2022
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-10: 103210158X
  • ISBN-13: 9781032101583
Kitos knygos pagal šią temą:
Galois Theory 5th editionDidesnis paveikslėlis
  • Formatas: Paperback / softback, 386 pages, aukštis x plotis: 234x156 mm, weight: 600 g, 38 Line drawings, black and white; 38 Illustrations, black and white
  • Išleidimo metai: 07-Sep-2022
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-10: 103210158X
  • ISBN-13: 9781032101583
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781032101583.html
Raktažodžiai:
Since 1973, Galois theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for todays algebra students.

New to the Fifth Edition





Reorganised and revised Chapters 7 and 13 New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations

This bestseller continues to deliver a rigorous, yet engaging, treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.

Recenzijos

"In mathematics, the fundamental theorem of Galois theory connects field theory and group theory, enabling certain mathematical problems in field theory to be reduced to group theory, making the problems simpler and easier to understand. The fifth updated edition of the textbook Galois Theory is an invaluable teaching text and resource for instructors of undergraduate mathematics students. Featuring more than 200 exercises and historical notes to enhance understanding of the proofs, formulas, and theorems, the fifth edition of Galois Theory is a "must-have" for university library mathematics collections, and highly recommended for instructors or for self-study" - Midwest Books Review

Praise for the Previous Editions" this book remains a highly recommended introduction to Galois theory along the more classical lines. It contains many exercises and a wealth of examples, including a pretty application of finite fields to the game solitaire. provides readers with insight and historical perspective; it is written for readers who would like to understand this central part of basic algebra rather than for those whose only aim is collecting credit points." Zentralblatt MATH 1322

"This edition preserves and even extends one of the most popular features of the original edition: the historical introduction and the story of the fatal duel of Evariste Galois. These historical notes should be of interest to students as well as mathematicians in general. after more than 30 years, Ian Stewarts Galois Theory remains a valuable textbook for algebra undergraduate students." Zentralblatt MATH, 1049

"The penultimate chapter is about algebraically closed fields and the last chapter, on transcendental numbers, contains what-every-mathematician-should-see-at-least-once, the proof of transcendence of pi. The book is designed for second- and third-year undergraduate courses. I will certainly use it." EMS Newsletter

Acknowledgements xiii
Preface to the Fifth Edition xv
Historical Introduction xix
1 Classical Algebra
1(18)
1.1 Complex Numbers
2(1)
1.2 Subfields and Subrings of the Complex Numbers
3(4)
1.3 Solving Equations
7(1)
1.4 Solution by Radicals
8(11)
2 The Fundamental Theorem of Algebra
19(12)
2.1 Polynomials
19(3)
2.2 Fundamental Theorem of Algebra
22(4)
2.3 Implications
26(5)
3 Factorisation of Polynomials
31(18)
3.1 The Euclidean Algorithm
31(5)
3.2 Irreducibility
36(2)
3.3 Gauss's Lemma
38(2)
3.4 Eisenstein's Criterion
40(2)
3.5 Reduction Modulo p
42(1)
3.6 Zeros of Polynomials
43(6)
4 Field Extensions
49(10)
4.1 Field Extensions
49(4)
4.2 Rational Expressions
53(1)
4.3 Simple Extensions
54(5)
5 Simple Extensions
59(10)
5.1 Algebraic and Transcendental Extensions
59(1)
5.2 The Minimal Polynomial
60(2)
5.3 Simple Algebraic Extensions
62(2)
5.4 Classifying Simple Extensions
64(5)
6 The Degree of an Extension
69(10)
6.1 Definition of the Degree
69(1)
6.2 The Tower Law
70(4)
6.3 Primitive Element Theorem
74(5)
7 Ruler-and-Compass Constructions
79(22)
7.1 Approximate Constructions and More General Instruments
81(2)
7.2 Constructions in C
83(5)
7.3 Specific Constructions
88(4)
7.4 Impossibility Proofs
92(2)
7.5 Construction from a Given Set of Points
94(7)
8 The Idea behind Galois Theory
101(22)
8.1 A First Look at Galois Theory
102(1)
8.2 Galois Groups According to Galois
102(3)
8.3 How to Use the Galois Group
105(1)
8.4 The Abstract Setting
106(1)
8.5 Polynomials and Extensions
107(2)
8.6 The Galois Correspondence
109(2)
8.7 Diet Galois
111(5)
8.8 Natural Irrationalities
116(7)
9 Normality and Separability
123(8)
9.1 Splitting Fields
123(3)
9.2 Normality
126(1)
9.3 Separability
127(4)
10 Counting Principles
131(8)
10.1 Linear Independence of Monomorphisms
131(8)
11 Field Automorphisms
139(6)
11.1 K-Monomorphisms
139(1)
11.2 Normal Closures
140(5)
12 The Galois Correspondence
145(4)
12.1 The Fundamental Theorem of Galois Theory
145(4)
13 Worked Examples
149(12)
13.1 Examples of Galois Groups
149(9)
13.2 Discussion
158(3)
14 Solubility and Simplicity
161(10)
14.1 Soluble Groups
161(3)
14.2 Simple Groups
164(3)
14.3 Cauchy's Theorem
167(4)
15 Solution by Radicals
171(10)
15.1 Radical Extensions
171(5)
15.2 An Insoluble Quintic
176(2)
15.3 Other Methods
178(3)
16 Abstract Rings and Fields
181(14)
16.1 Rings and Fields
182(2)
16.2 General Properties of Rings and Fields
184(2)
16.3 Polynomials Over General Rings
186(1)
16.4 The Characteristic of a Field
187(2)
16.5 Integral Domains
189(6)
17 Abstract Field Extensions and Galois Groups
195(20)
17.1 Minimal Polynomials
195(1)
17.2 Simple Algebraic Extensions
196(1)
17.3 Splitting Fields
197(2)
17.4 Normality
199(1)
17.5 Separability
200(4)
17.6 Galois Theory for Abstract Fields
204(1)
17.7 Conjugates and Minimal Polynomials
205(3)
17.8 The Primitive Element Theorem
208(1)
17.9 Algebraic Closure of a Field
209(6)
18 The General Polynomial Equation
215(18)
18.1 Transcendence Degree
215(3)
18.2 Elementary Symmetric Polynomials
218(2)
18.3 The General Polynomial
220(1)
18.4 Cyclic Extensions
221(4)
18.5 Solving Equations of Degree Four or Less
225(2)
18.6 Explicit Formulas
227(6)
19 Finite Fields
233(8)
19.1 Structure of Finite Fields
233(1)
19.2 The Multiplicative Group
234(2)
19.3 Counterexample to the Primitive Element Theorem
236(1)
19.4 Application to Solitaire
237(4)
20 Regular Polygons
241(16)
20.1 What Euclid Knew
241(3)
20.2 Which Constructions are Possible?
244(1)
20.3 Regular Polygons
245(4)
20.4 Fermat Numbers
249(1)
20.5 How to Construct a Regular 17-gon
249(8)
21 Circle Division
257(26)
21.1 Genuine Radicals
258(2)
21.2 Fifth Roots Revisited
260(3)
21.3 Vandermonde Revisited
263(2)
21.4 The General Case
265(3)
21.5 Cyclotomic Polynomials
268(3)
21.6 Galois Group of Q(ζ)/Q
271(3)
21.7 Constructions Using a Trisector
274(9)
22 Calculating Galois Groups
283(10)
22.1 Transitive Subgroups
283(2)
22.2 Bare Hands on the Cubic
285(2)
22.3 The Discriminant
287(1)
22.4 General Algorithm for the Galois Group
288(5)
23 Algebraically Closed Fields
293(8)
23.1 Ordered Fields and Their Extensions
293(2)
23.2 Sylow's Theorem
295(2)
23.3 The Algebraic Proof
297(4)
24 Transcendental Numbers
301(10)
24.1 Irrationality
302(2)
24.2 Transcendence of e
304(1)
24.3 Transcendence of π
305(6)
25 What Did Galois Do or Know?
311(16)
25.1 List of the Relevant Material
312(1)
25.2 The First Memoir
312(1)
25.3 What Galois Proved
313(2)
25.4 What is Galois up to?
315(2)
25.5 Alternating Groups, Especially A5
317(1)
25.6 Simple Groups Known to Galois
318(1)
25.7 Speculations about Proofs
319(5)
25.8 A5 is Unique
324(3)
26 Further Directions
327(10)
26.1 Inverse Galois Problem
327(2)
26.2 Differential Galois Theory
329(4)
26.3 p-adic Numbers
333(4)
References 337(8)
Index 345
Ian Stewart is an emeritus professor of mathematics at the University of Warwick and a fellow of the Royal Society. Dr. Stewart has been a recipient of many honors, including the Royal Societys Faraday Medal, the IMA Gold Medal, the AAAS Public Understanding of Science and Technology Award, and the LMS/IMA Zeeman Medal. He has published more than 210 scientific papers and numerous books, including several bestsellers co-authored with Terry Pratchett and Jack Cohen that combine fantasy with nonfiction.