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176 Pages
176 Pages
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22.86 x 15.24 x 1.04
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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.
Industry Reviews
"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
Preface | p. vii |
Algebraic preliminaries | p. 1 |
Groups, fields and vector spaces | p. 3 |
Groups | p. 3 |
Fields | p. 8 |
Vector spaces | p. 9 |
The axiom of choice, and Zorn's lemma | p. 14 |
The axiom of choice | p. 14 |
Zorn's lemma | p. 14 |
The existence of a basis | p. 15 |
Rings | p. 18 |
Rings | p. 18 |
Integral domains | p. 20 |
Ideals | p. 21 |
Irreducibles, primes and unique factorization domains | p. 24 |
Principal ideal domains | p. 27 |
Highest common factors | p. 29 |
Polynomials over unique factorization domains | p. 31 |
The existence of maximal proper ideals | p. 34 |
More about fields | p. 35 |
The theory of fields, and Galois theory | p. 37 |
Field extensions | p. 39 |
Introduction | p. 39 |
Field extensions | p. 40 |
Algebraic and transcendental elements | p. 42 |
Algebraic extensions | p. 46 |
Monomorphisms of algebraic extensions | p. 48 |
Tests for irreducibility | p. 49 |
Introduction | p. 49 |
Eisenstein's criterion | p. 51 |
Other methods for establishing irreducibility | p. 52 |
Ruler-and-compass constructions | p. 54 |
Constructible points | p. 54 |
The angle [pi]/3 cannot be trisected | p. 57 |
Concluding remarks | p. 58 |
Splitting fields | p. 59 |
Splitting fields | p. 60 |
The extension of monomorphisms | p. 62 |
Some examples | p. 67 |
The algebraic closure of a field | p. 71 |
Introduction | p. 71 |
The existence of an algebraic closure | p. 72 |
The uniqueness of an algebraic closure | p. 75 |
Conclusions | p. 77 |
Normal extensions | p. 78 |
Basic properties | p. 78 |
Monomorphisms and automorphisms | p. 80 |
Separability | p. 82 |
Basic ideas | p. 82 |
Monomorphisms and automorphisms | p. 83 |
Galois extensions | p. 84 |
Differentiation | p. 85 |
The Frobenius monomorphism | p. 87 |
Inseparable polynomials | p. 88 |
Automorphisms and fixed fields | p. 91 |
Fixed fields and Galois groups | p. 91 |
The Galois group of a polynomial | p. 94 |
An example | p. 96 |
The fundamental theorem of Galois theory | p. 97 |
The theorem on natural irrationalities | p. 99 |
Finite fields | p. 101 |
A description of the finite fields | p. 101 |
An example | p. 102 |
Some abelian group theory | p. 103 |
The multiplicative group of a finite field | p. 105 |
The automorphism group of a finite field | p. 105 |
The theorem of the primitive element | p. 107 |
A criterion in terms of intermediate fields | p. 107 |
The theorem of the primitive element | p. 108 |
An example | p. 109 |
Cubics and quartics | p. 110 |
Extension by radicals | p. 110 |
The discriminant | p. 111 |
Cubic polynomials | p. 113 |
Quartic polynomials | p. 115 |
Roots of unity | p. 118 |
Cyclotomic polynomials | p. 118 |
Irreducibility | p. 120 |
The Galois group of a cyclotomic polynomial | p. 121 |
Cyclic extensions | p. 123 |
A necessary condition | p. 123 |
Abel's theorem | p. 124 |
A sufficient condition | p. 125 |
Kummer extensions | p. 128 |
Solution by radicals | p. 131 |
Soluble groups: examples | p. 131 |
Soluble groups: basic theory | p. 132 |
Polynomials with soluble Galois groups | p. 134 |
Polynomials which are solvable by radicals | p. 135 |
Transcendental elements and algebraic independence | p. 139 |
Transcendental elements and algebraic independence | p. 139 |
Transcendence bases | p. 141 |
Transcendence degree | p. 143 |
The tower law for transcendence degree | p. 144 |
Luroth's theorem | p. 145 |
Some further topics | p. 147 |
Generic polynomials | p. 147 |
The normal basis theorem | p. 150 |
Constructing regular polygons | p. 152 |
The calculation of Galois groups | p. 155 |
A procedure for determining the Galois group of a polynomial | p. 155 |
The soluble transitive subgroups of [Sigma subscript p] | p. 158 |
The Galois group of a quintic | p. 161 |
Concluding remarks | p. 162 |
Index | p. 163 |
Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780521312493
ISBN-10: 0521312493
Published: 1st August 1987
Format: Paperback
Language: English
Number of Pages: 176
Audience: College, Tertiary and University
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24 x 1.04
Weight (kg): 0.22
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