You have reached 400 XP and carrot coins. That is the daily max!

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 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… More 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.Less

Customers also bought

Book details

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: 0.484 Language: English

AuthorTable of Contents

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

Free shipping

A minimum purchase of $35 is required. Shipping is provided via FedEx SmartPost® and FedEx Express Saver®. Average delivery time is 1 – 5 business days, but is not guaranteed in that timeframe. Also allow 1 - 2 days for processing. Free shipping is eligible only in the continental United States and excludes Hawaii, Alaska and Puerto Rico. FedEx service marks used by permission."Marketplace" orders are not eligible for free or discounted shipping.

Our guarantee

If an item you ordered from TextbookRush does not meet your expectations due to an error on our part, simply fill out a return request and then return it by mail within 30 days of ordering it for a full refund of item cost.