Course in Galois Theory

ISBN-10: 0521312493
ISBN-13: 9780521312493
Edition: 1986
Authors: D. J. H. Garling
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Description: 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  More...

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Book details

List price: $49.99
Copyright year: 1986
Publisher: Cambridge University Press
Publication date: 1/8/1987
Binding: Paperback
Pages: 176
Size: 6.00" wide x 9.00" long x 0.25" tall
Weight: 0.814
Language: English

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.

D. J. H. Garling is a Fellow of St John's College and Emeritus Reader in Mathematical Analysis at the University of Cambridge, in the Department of Pure Mathematics and Mathematical Statistics.

Preface
Algebraic preliminaries
Groups, fields and vector spaces
Groups
Fields
Vector spaces
The axiom of choice, and Zorn's lemma
The axiom of choice
Zorn's lemma
The existence of a basis
Rings
Rings
Integral domains
Ideals
Irreducibles, primes and unique factorization domains
Principal ideal domains
Highest common factors
Polynomials over unique factorization domains
The existence of maximal proper ideals
More about fields
The theory of fields, and Galois theory
Field extensions
Introduction
Field extensions
Algebraic and transcendental elements
Algebraic extensions
Monomorphisms of algebraic extensions
Tests for irreducibility
Introduction
Eisenstein's criterion
Other methods for establishing irreducibility
Ruler-and-compass constructions
Constructible points
The angle [pi]/3 cannot be trisected
Concluding remarks
Splitting fields
Splitting fields
The extension of monomorphisms
Some examples
The algebraic closure of a field
Introduction
The existence of an algebraic closure
The uniqueness of an algebraic closure
Conclusions
Normal extensions
Basic properties
Monomorphisms and automorphisms
Separability
Basic ideas
Monomorphisms and automorphisms
Galois extensions
Differentiation
The Frobenius monomorphism
Inseparable polynomials
Automorphisms and fixed fields
Fixed fields and Galois groups
The Galois group of a polynomial
An example
The fundamental theorem of Galois theory
The theorem on natural irrationalities
Finite fields
A description of the finite fields
An example
Some abelian group theory
The multiplicative group of a finite field
The automorphism group of a finite field
The theorem of the primitive element
A criterion in terms of intermediate fields
The theorem of the primitive element
An example
Cubics and quartics
Extension by radicals
The discriminant
Cubic polynomials
Quartic polynomials
Roots of unity
Cyclotomic polynomials
Irreducibility
The Galois group of a cyclotomic polynomial
Cyclic extensions
A necessary condition
Abel's theorem
A sufficient condition
Kummer extensions
Solution by radicals
Soluble groups: examples
Soluble groups: basic theory
Polynomials with soluble Galois groups
Polynomials which are solvable by radicals
Transcendental elements and algebraic independence
Transcendental elements and algebraic independence
Transcendence bases
Transcendence degree
The tower law for transcendence degree
Luroth's theorem
Some further topics
Generic polynomials
The normal basis theorem
Constructing regular polygons
The calculation of Galois groups
A procedure for determining the Galois group of a polynomial
The soluble transitive subgroups of [Sigma subscript p]
The Galois group of a quintic
Concluding remarks
Index

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