
Book DetailsThe Carus Mathematical MonographsVolume: 19; 1975; 323 ppMSC: Primary 12
Reprinted edition available: CAR/35
Field Theory and its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the construction of regular \(n\)gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals and beyond. The logical pathway is historic, but the terminology is consistent with modern treatments. No previous knowledge of algebra is assumed. Notable topics treated along this route include the transcendence of \(e\) and \(\pi\), cyclotomic polynomials, polynomials over the integers, Hilbert's irreducibility theorem, and many other gems in classical mathematics. Historical and bibliographical notes complement the text, and complete solutions are provided to all problems.

Table of Contents

Front Cover

Field Theory and its Classical Problems

Copyright Page

Preface

Contents

Introduction

Chapter 1—The Three Greek Problems

1.1. Constructible Lengths

1.2. Doubling the Cube

1 3 Trisecting the Angle

1.4. Squaring the Circle

1.5. Polynomials and Their Roots

1.6. Symmetric Functions

1.7. The Transcendence of π

References and Notes

Chapter 2—Field Extensions

2.1. Arithmetic of Polynomials

2.2. Simple, Multiple, and Finite Extensions

2.3. Geometric Constructions Revisited

2.4. Roots of Complex Numbers

2.5. Constructibility of Regular Polygons I

2.6. Congruences

2.7. Constructibility of Regular Polygons II

References and Notes

Chapter 3—Solution by Radicals

3.1. Statement of the Problem

3.2. Automorphisms and Groups

3.3. The Group of an Extension

3.4. Two Fundamental Theorems

3.5. Galois' Theorem

3.6. Abel's Theorem

3.7. Some Solvable Equations

References and Notes

Chapter 4—Polynomials with Symmetric Groups

4.1. Background Information

4.2. Hubert's Irreducibility Theorem

4.3. Existence of Polynomials over Q with Group Sn

References and Notes

Solutions to the Problems

Index


Additional Material

Reviews

The presented book is a clear and concise introduction to classical results of Galois theory. The book is an excellent reading for everyone, especially for instructors and first year graduate students in Galois theory.
Acta. Sci. Math.


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Reprinted edition available: CAR/35
Field Theory and its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the construction of regular \(n\)gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals and beyond. The logical pathway is historic, but the terminology is consistent with modern treatments. No previous knowledge of algebra is assumed. Notable topics treated along this route include the transcendence of \(e\) and \(\pi\), cyclotomic polynomials, polynomials over the integers, Hilbert's irreducibility theorem, and many other gems in classical mathematics. Historical and bibliographical notes complement the text, and complete solutions are provided to all problems.

Front Cover

Field Theory and its Classical Problems

Copyright Page

Preface

Contents

Introduction

Chapter 1—The Three Greek Problems

1.1. Constructible Lengths

1.2. Doubling the Cube

1 3 Trisecting the Angle

1.4. Squaring the Circle

1.5. Polynomials and Their Roots

1.6. Symmetric Functions

1.7. The Transcendence of π

References and Notes

Chapter 2—Field Extensions

2.1. Arithmetic of Polynomials

2.2. Simple, Multiple, and Finite Extensions

2.3. Geometric Constructions Revisited

2.4. Roots of Complex Numbers

2.5. Constructibility of Regular Polygons I

2.6. Congruences

2.7. Constructibility of Regular Polygons II

References and Notes

Chapter 3—Solution by Radicals

3.1. Statement of the Problem

3.2. Automorphisms and Groups

3.3. The Group of an Extension

3.4. Two Fundamental Theorems

3.5. Galois' Theorem

3.6. Abel's Theorem

3.7. Some Solvable Equations

References and Notes

Chapter 4—Polynomials with Symmetric Groups

4.1. Background Information

4.2. Hubert's Irreducibility Theorem

4.3. Existence of Polynomials over Q with Group Sn

References and Notes

Solutions to the Problems

Index

The presented book is a clear and concise introduction to classical results of Galois theory. The book is an excellent reading for everyone, especially for instructors and first year graduate students in Galois theory.
Acta. Sci. Math.