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A Course in Galois Theory - D. J. H. Garling

A Course in Galois Theory


Published: 9th March 1987
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Galois theory is one of the most beautiful branches of mathematics. By synthesising the techniques of group theory and field theory it provides a complete answer to the problem of the solubility of polynomials by radicals: that is, the problem of determining when and how a polynomial equation can be solved by repeatedly extracting roots and using elementary algebraic operations. This textbook, based on lectures given over a period of years at Cambridge, is a detailed and thorough introduction to the subject. The work begins with an elementary discussion of groups, fields and vector spaces, and then leads the reader through such topics as rings, extension fields, ruler-and-compass constructions, to automorphisms and the Galois correspondence. By these means, the problem of the solubility of polynomials by radicals is answered; in particular it is shown that not every quintic equation can be solved by radicals. Throughout, Dr Garling presents the subject not as something closed, but as one with many applications. In the final chapters, he discusses further topics, such as transcendence and the calculation of Galois groups, which indicate that there are many questions still to be answered. The reader is assumed to have no previous knowledge of Galois theory. Some experience of modern algebra is helpful, so that the book is suitable for undergraduates in their second or final years. There are over 200 exercises which provide a stimulating challenge to the reader.

"This is a marvellous little book. It is characterized by good mathematical taste, plain and elegant language, and an earthy but precise style." Carl Riehm, Mathematical Reviews

Prefacep. vii
Algebraic preliminariesp. 1
Groups, fields and vector spacesp. 3
Groupsp. 3
Fieldsp. 8
Vector spacesp. 9
The axiom of choice, and Zorn's lemmap. 14
The axiom of choicep. 14
Zorn's lemmap. 14
The existence of a basisp. 15
Ringsp. 18
Ringsp. 18
Integral domainsp. 20
Idealsp. 21
Irreducibles, primes and unique factorization domainsp. 24
Principal ideal domainsp. 27
Highest common factorsp. 29
Polynomials over unique factorization domainsp. 31
The existence of maximal proper idealsp. 34
More about fieldsp. 35
The theory of fields, and Galois theoryp. 37
Field extensionsp. 39
Introductionp. 39
Field extensionsp. 40
Algebraic and transcendental elementsp. 42
Algebraic extensionsp. 46
Monomorphisms of algebraic extensionsp. 48
Tests for irreducibilityp. 49
Introductionp. 49
Eisenstein's criterionp. 51
Other methods for establishing irreducibilityp. 52
Ruler-and-compass constructionsp. 54
Constructible pointsp. 54
The angle [pi]/3 cannot be trisectedp. 57
Concluding remarksp. 58
Splitting fieldsp. 59
Splitting fieldsp. 60
The extension of monomorphismsp. 62
Some examplesp. 67
The algebraic closure of a fieldp. 71
Introductionp. 71
The existence of an algebraic closurep. 72
The uniqueness of an algebraic closurep. 75
Conclusionsp. 77
Normal extensionsp. 78
Basic propertiesp. 78
Monomorphisms and automorphismsp. 80
Separabilityp. 82
Basic ideasp. 82
Monomorphisms and automorphismsp. 83
Galois extensionsp. 84
Differentiationp. 85
The Frobenius monomorphismp. 87
Inseparable polynomialsp. 88
Automorphisms and fixed fieldsp. 91
Fixed fields and Galois groupsp. 91
The Galois group of a polynomialp. 94
An examplep. 96
The fundamental theorem of Galois theoryp. 97
The theorem on natural irrationalitiesp. 99
Finite fieldsp. 101
A description of the finite fieldsp. 101
An examplep. 102
Some abelian group theoryp. 103
The multiplicative group of a finite fieldp. 105
The automorphism group of a finite fieldp. 105
The theorem of the primitive elementp. 107
A criterion in terms of intermediate fieldsp. 107
The theorem of the primitive elementp. 108
An examplep. 109
Cubics and quarticsp. 110
Extension by radicalsp. 110
The discriminantp. 111
Cubic polynomialsp. 113
Quartic polynomialsp. 115
Roots of unityp. 118
Cyclotomic polynomialsp. 118
Irreducibilityp. 120
The Galois group of a cyclotomic polynomialp. 121
Cyclic extensionsp. 123
A necessary conditionp. 123
Abel's theoremp. 124
A sufficient conditionp. 125
Kummer extensionsp. 128
Solution by radicalsp. 131
Soluble groups: examplesp. 131
Soluble groups: basic theoryp. 132
Polynomials with soluble Galois groupsp. 134
Polynomials which are solvable by radicalsp. 135
Transcendental elements and algebraic independencep. 139
Transcendental elements and algebraic independencep. 139
Transcendence basesp. 141
Transcendence degreep. 143
The tower law for transcendence degreep. 144
Luroth's theoremp. 145
Some further topicsp. 147
Generic polynomialsp. 147
The normal basis theoremp. 150
Constructing regular polygonsp. 152
The calculation of Galois groupsp. 155
A procedure for determining the Galois group of a polynomialp. 155
The soluble transitive subgroups of [Sigma subscript p]p. 158
The Galois group of a quinticp. 161
Concluding remarksp. 162
Indexp. 163
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521312493
ISBN-10: 0521312493
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 176
Published: 9th March 1987
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.1
Weight (kg): 0.22