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Galois Theory and Its Algebraic Background 2nd Revised edition [Mīkstie vāki]

(University of Cambridge)
  • Formāts: Paperback / softback, 204 pages, height x width x depth: 229x151x18 mm, weight: 360 g, Worked examples or Exercises
  • Izdošanas datums: 22-Jul-2021
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1108969089
  • ISBN-13: 9781108969086
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Galois Theory and Its Algebraic Background 2nd Revised editionPalielināt
  • Formāts: Paperback / softback, 204 pages, height x width x depth: 229x151x18 mm, weight: 360 g, Worked examples or Exercises
  • Izdošanas datums: 22-Jul-2021
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1108969089
  • ISBN-13: 9781108969086
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781108969086.html
Keywords:
Galois Theory is the theory of polynomial equations and their solutions. Suitable for course-following undergraduates and the independent reader, this textbook gives a full account of Galois Theory and the necessary background algebra. This second edition has been revised and re-ordered, with new exercises and examples throughout.

Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.

Recenzijas

'Garling's book presents Galois theory in a style which is at once readable and compact. The necessary prerequisites are developed in the early chapters only to the extent that they are needed later. The proofs of the lemmas and main theorems are presented in as concrete a manner as possible, without unnecessary abstraction. Yet they seem remarkably short, without the difficulties being glossed over. In fact the approach throughout the book is down-to-earth and concrete I can heartily recommend this book as an undergraduate text.' Bulletin of the London Mathematical Society

Papildus informācija

This textbook contains a full account of Galois Theory and the algebra that it needs, with exercises, examples and applications.
Preface ix
PART I THE ALGEBRAIC BACKGROUND
1 Groups
3(22)
1.1 Groups
3(6)
1.2 Finite Abelian Groups
9(2)
1.3 Finite Permutation Groups
11(6)
1.4 Group Series
17(1)
1.5 Soluble Groups
18(4)
1.6 p-Groups and Sylow Theorems
22(3)
2 Integral Domains
25(29)
2.1 Commutative Rings with 1
25(1)
2.2 Polynomials
26(1)
2.3 Homomorphisms and Ideals
27(2)
2.4 Integral Domains
29(2)
2.5 Fields and Fractions
31(4)
2.6 The Ordered Set of Ideals in an Integral Domain
35(1)
2.7 Factorization
36(2)
2.8 Unique Factorization
38(3)
2.9 Principal Ideal Domains and Euclidean Domains
41(3)
2.10 Polynomials Over Unique Factorization Domains
44(3)
2.11 More About Fields
47(1)
2.12 Kronecker's Algorithm
48(2)
2.13 Eisenstein's Criterion
50(1)
2.14 Localization
51(3)
3 Vector Spaces and Determinants
54(15)
3.1 Vector Spaces
54(6)
3.2 The Infinite-Dimensional Case
60(1)
3.3 Characters and Automorphisms
61(1)
3.4 Determinants
62(7)
PART II THE THEORY OF FIELDS AND GALOIS THEORY
4 Field Extensions
69(12)
4.1 Introduction
69(1)
4.2 Field Extensions
70(3)
4.3 Algebraic and Transcendental Extensions
73(4)
4.4 Algebraic Extensions
77(3)
4.5 Monomorphisms of Algebraic Extensions
80(1)
5 Ruler and Compass Constructions
81(4)
5.1 Some Classical Problems
81(1)
5.2 Constructible Points
81(4)
6 Splitting Fields
85(13)
6.1 Introduction
85(1)
6.2 Splitting Fields
86(3)
6.3 The Extension of Monomorphisms
89(5)
6.4 Some Examples
94(4)
7 Normal Extensions
98(5)
7.1 Basic Properties
98(3)
7.2 Monomorphisms and Automorphisms
101(2)
8 Separability
103(9)
8.1 Basic Ideas
103(1)
8.2 Monomorphisms and Automorphisms
104(2)
8.3 Galois Extensions
106(1)
8.4 Differentiation
106(2)
8.5 Inseparable Polynomials
108(4)
9 The Fundamental Theorem of Galois Theory
112(10)
9.1 Field Automorphisms, Fixed Fields and Galois Groups
112(1)
9.2 Linear Independence
113(2)
9.3 The Size of a Galois Group is the Degree of the Extension
115(1)
9.4 The Galois Group of a Polynomial
116(2)
9.5 The Fundamental Theorem of Galois Theory
118(4)
10 The Discriminant
122(4)
10.1 The Discriminant
122(4)
11 Cyclotomic Polynomials and Cyclic Extensions
126(14)
11.1 Cyclotomic Polynomials
126(2)
11.2 Irreducibility
128(1)
11.3 The Galois Group of a Cyclotomic Polynomial
129(2)
11.4 A Necessary Condition
131(1)
11.5 Abel's Theorem
132(2)
11.6 Norms and Traces
134(1)
11.7 A Sufficient Condition
135(2)
11.8 Kummer Extensions
137(3)
12 Solution by Radicals
140(6)
12.1 Polynomials with Soluble Galois Groups
140(1)
12.2 Polynomials which are Soluble by Radicals
141(5)
13 Regular Polygons
146(3)
13.1 Fermat Primes and Fermat Numbers
146(1)
13.2 Regular Polygons
147(1)
13.3 Constructing a Regular Pentagon
148(1)
14 Polynomials of Low Degree
149(7)
14.1 Quadratic Polynomials
149(1)
14.2 Cubic Polynomials
150(3)
14.3 Quartic Polynomials
153(3)
15 Finite Fields
156(5)
15.1 Finite Fields
156(1)
15.2 Polynomials in Zp[ x]
157(1)
15.3 Polynomials of Low Degree over a Finite Field
158(3)
16 Quintic Polynomials
161(3)
17 Further Theory
164(6)
17.1 Simple Extensions
165(1)
17.2 The Theorem of the Primitive Element
166(2)
17.3 The Normal Basis Theorem
168(2)
18 The Algebraic Closure of a Field
170(7)
18.1 Introduction
170(1)
18.2 The Existence of an Algebraic Closure
171(4)
18.3 The Uniqueness of an Algebraic Closure
175(1)
18.4 Conclusions
176(1)
19 Transcendental Elements and Algebraic Independence
177(9)
19.1 Transcendental Elements and Algebraic Independence
177(3)
19.2 Transcendence Bases
180(1)
19.3 Transcendence Degree
181(1)
19.4 The Tower Law for Transcendence Degree
182(1)
19.5 Liiroth's Theorem
183(3)
20 Generic and Symmetric Polynomials
186(3)
20.1 Generic and Symmetric Polynomials
186(3)
Appendix: The Axiom of Choice 189(3)
Index 192
D. J. H. Garling is Emeritus Reader in Mathematical Analysis at the University of Cambridge and Fellow of St John's College, Cambridge. He has fifty years' experience of teaching undergraduate students and has written several books on mathematics, including Inequalities: A Journey into Linear Analysis (Cambridge, 2007) and A Course in Mathematical Analysis (Three volumes, Cambridge, 20132014).