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Galois Theory 2001 ed. [Kietas viršelis]

3.50/5 (11 ratings by Goodreads)

Translated by L. Schneps, Jean-Pierre Escofier

Formatas: Hardback, 283 pages, aukštis x plotis: 235x155 mm, weight: 1330 g, 28 Tables, black and white; XIV, 283 p., 1 Hardback
Serija: Graduate Texts in Mathematics 204
Išleidimo metai: 21-Dec-2000
Leidėjas: Springer-Verlag New York Inc.
ISBN-10: 0387987657
ISBN-13: 9780387987651

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Algebra - (Šiuo metu turimos knygos: 3 prekės)
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Biography: general - (Šiuo metu turimos knygos: 66 prekės)

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Formatas: Hardback, 283 pages, aukštis x plotis: 235x155 mm, weight: 1330 g, 28 Tables, black and white; XIV, 283 p., 1 Hardback
Serija: Graduate Texts in Mathematics 204
Išleidimo metai: 21-Dec-2000
Leidėjas: Springer-Verlag New York Inc.
ISBN-10: 0387987657
ISBN-13: 9780387987651

Kitos knygos pagal šią temą:

Algebra - (Šiuo metu turimos knygos: 3 prekės)
Number theory - (Šiuo metu turimos knygos: 1 prekės)
Biography: general - (Šiuo metu turimos knygos: 66 prekės)

Pastovi nuoroda: https://www.kriso.lt/db/9780387987651.html

Raktažodžiai:

This book offers the fundamentals of Galois Theory, including a set of copious, well-chosen exercises that form an important part of the presentation. The pace is gentle and incorporates interesting historical material, including aspects on the life of Galois. Computed examples, recent developments, and extensions of results into other related areas round out the presentation.

Recenzijos

J.-P. Escofier

Galois Theory

"Escofiers treatment, at a level suitable for advanced, senior undergraduates or first-year graduate students, centers on finite extensions of number fields, incorporating numerous examples and leaving aside finite fields and the entire concept of separability for the final chapters . . . copious, well-chosen exercises . . . are presented with their solutions . . . The prose is . . . spare and enthusiastic, and the proofs are both instructive and efficient . . . Escofier has written an excellent text, offering a relatively elementary introduction to a beautiful subject in a book sufficiently broad to present a contemporary viewpoint and intuition but sufficiently restrained so as not to overwhelm the reader."AMERICAN MATHEMATICAL SOCIETY

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Springer Book Archives

Preface

Historical Aspects of the Resolution of Algebraic Equations

(8)

Approximating the Roots of an Equation

(1)

Construction of Solutions by Intersections of Curves

(1)

Relations with Trigonometry

(1)

Problems of Notation and Terminology

(1)

The Problem of Localization of the Roots

(1)

The Problem of the Existence of Roots

(1)

The Problem of Algebraic Solutions of Equations

(3)

Toward
Chapter 2

(2)

Resolution of Quadratic, Cubic, and Quartic Equations

(16)

Second-Degree Equations

(4)

The Babylonians

(2)

The Greeks

(1)

The Arabs

(1)

Use of Negative Numbers

(1)

Cubic Equations

(5)

The Greeks

(1)

Omar Khayyam and Sharaf ad Din at Tusi

(1)

Scipio del Ferro, Tartaglia, Cardan

(1)

Algebraic Solution of the Cubic Equation

(1)

First Computations with Complex Numbers

(1)

Raffaele Bombelli

(1)

Francois Viete

(1)

Quartic Equations

(7)

Exercises for
Chapter 2

(3)

Solutions to Some of the Exercises

(3)

Symmetric Polynomials

(26)

Symmetric Polynomials

(2)

Background

(1)

Definitions

(1)

Elementary Symmetric Polynomials

(2)

Definition

(1)

The Product of the X - Xi; Relations Between Coefficients and Roots

(2)

Symmetric Polynomials and Elementary Symmetric Polynomials

(3)

Theorem

(2)

Proposition

(1)

Proposition

(1)

Newton's Formulas

(3)

Resultant of Two Polynomials

(2)

Definition

(1)

Proposition

(2)

Discriminant of a Polynomial

(14)

Definition

(1)

Proposition

(1)

Formulas

(1)

Polynomials with Real Coefficients: Real Roots and Sign of the Discriminant

(1)

Exercises for
Chapter 3

(5)

Solutions to Some of the Exercises

(7)

Field Extensions

(28)

Field Extensions

(2)

Definition

(1)

Proposition

(1)

The Degree of an Extension

(1)

Towers of Fields

(1)

The Tower Rule

(1)

Proposition

(1)

Generated Extensions

(1)

Proposition

(1)

Definition

(1)

Proposition

(1)

Algebraic Elements

(4)

Definition

(1)

Transcendental Numbers

(1)

Minimal Polynomial of an Algebraic Element

(1)

Definition

(1)

Properties of the Minimal Polynomial

(1)

Proving the Irreducibility of a Polynomial in Z[ X]

(2)

Algebraic Extensions

(2)

Extensions Generated by an Algebraic Element

(1)

Properties of K[ a]

(1)

Definition

(1)

Extensions of Finite Degree

(1)

Corollary: Towers of Algebraic Extensions

(1)

Algebraic Extensions Generated by n Elements

(1)

Notation

(1)

Proposition

(1)

Corollary

(1)

Construction of an Extension by Adjoining a Root

(17)

Definition

(1)

Proposition

(1)

Corollary

(1)

Universal Property of K[ X]/(P)

(1)

Toward
Chapters 5 and 6

(1)

Exercises for
Chapter 4

(5)

Solutions to Some of the Exercises

(10)

Constructions with Straightedge and Compass

(14)

Constructible Points

(1)

Examples of Classical Constructions

(1)

Projection of a Point onto a Line

(1)

Construction of an Orthonormal Basis from Two Points

(1)

Construction of a Line Parallel to a Given Line Passing Through a Point

(1)

Lemma

(1)

Coordinates of Points Constructible in One Step

(1)

A Necessary Condition for Constructibility

(1)

Two Problems More Than Two Thousand Years Old

(1)

Duplication of the Cube

(1)

Trisection of the Angle

(1)

A Sufficient Condition for Constructibility

(8)

Exercises for
Chapter 5

(3)

Solutions to Some of the Exercises

(3)

K-Homomorphisms

(14)

Conjugate Numbers

(1)

K-Homomorphisms

(1)

Definitions

(1)

Properties

(1)

Algebraic Elements and K-Homomorphisms

(2)

Proposition

(1)

Example

(1)

Extensions of Embeddings into C

(2)

Definition

(1)

Proposition

(1)

Proposition

(1)

The Primitive Element Theorem

(2)

Theorem and Definition

(1)

Example

100

(1)

Linear Independence of K-Homomorphisms

101

(6)

Characters

101

(1)

Emil Artin's Theorem

101

(1)

Corollary: Dedekind's Theorem

102

(1)

Exercises for
Chapter 6

102

(1)

Solutions to Some of the Exercises

103

(4)

Normal Extensions

107

(12)

Splitting Fields

107

(1)

Definition

107

(1)

Splitting Field of a Cubic Polynomial

108

(1)

Normal Extensions

108

(1)

Normal Extensions and K-Homomorphisms

109

(1)

Splitting Fields and Normal Extensions

109

(1)

Proposition

109

(1)

Converse

110

(1)

Normal Extensions and Intermediate Extensions

110

(1)

Normal Closure

111

(1)

Definition

111

(1)

Proposition

111

(1)

Proposition

111

(1)

Splitting Fields: General Case

112

(7)

Toward
Chapter 8

113

(1)

Exercises for
Chapter 7

113

(2)

Solutions to Some of the Exercises

115

(4)

Galois Groups

119

(30)

Galois Groups

119

(3)

The Galois Group of an Extension

119

(1)

The Order of the Galois Group of a Normal Extension of Finite Degree

120

(1)

The Galois Group of a Polynomial

120

(1)

The Galois Group as a Subgroup of a Permutation Group

120

(1)

A Short History of Groups

121

(1)

Fields of Invariants

122

(2)

Definition and Proposition

122

(1)

Emil Artin's Theorem

122

(2)

The Example of Q [ 3√2, j]: First Part

124

(2)

Galois Groups and Intermediate Extensions

126

(1)

The Galois Correspondence

126

(2)

The Example of Q [ 3√2, j]: Second Part

128

(1)

The Example X4 + 2

128

(21)

Dihedral Groups

128

(1)

The Special Case of D4

129

(1)

The Galois Group of X4 + 2

130

(1)

The Galois Correspondence

130

(2)

Search for Minimal Polynomials

132

(1)

Toward
Chapters 9, 10, and 12

133

(1)

Exercises for
Chapter 8

133

(6)

Solutions to Some of the Exercises

139

(10)

Roots of Unity

149

(30)

The Group U(n) of Units of the Ring Z/nZ

149

(2)

Definition and Background

149

(1)

The Structure of U(n)

150

(1)

The Mobius Function

151

(2)

Multiplicative Functions

151

(1)

The Mobius Function

151

(1)

Proposition

151

(1)

The Mobius Inversion Formula

152

(1)

Roots of Unity

153

(1)

n-th Roots of Unity

153

(1)

Proposition

153

(1)

Primitive Roots

153

(1)

Properties of Primitive Roots

153

(1)

Cyclotomic Polynomials

153

(3)

Definition

153

(1)

Properties of the Cyclotomic Polynomial

153

(3)

The Galois Group over Q of an Extension of Q by a Root of Unity

156

(23)

Exercises for
Chapter 9

157

(6)

Solutions to Some of the Exercises

163

(16)

Cyclic Extensions

179

(16)

Cyclic and Abelian Extensions

179

(1)

Extensions by a Root and Cyclic Extensions

179

(1)

Irreducibility of Xp - a

180

(1)

Hilbert's Theorem 90

181

(1)

The Norm

181

(1)

Hilbert's Theorem 90

182

(1)

Extensions by a Root and Cyclic Extensions: Converse

182

(1)

Lagrange Resolvents

183

(1)

Definition

183

(1)

Properties

183

(1)

Resolution of the Cubic Equation

184

(2)

Solution of the Quartic Equation

186

(2)

Historical Commentary

188

(7)

Exercises for
Chapter 10

188

(2)

Solutions to Some of the Exercises

190

(5)

Solvable Groups

195

(12)

First Definition

195

(1)

Derived or Commutator Subgroup

196

(1)

Second Definition of Solvability

196

(1)

Examples of Solvable Groups

197

(1)

Third Definition

197

(1)

The Group An Is Simple for n ≥ 5

198

(1)

Theorem

198

(1)

An Is Not Solvable for n ≥ 5, Direct Proof

199

(1)

Galois Theory 2001 ed. [Kietas viršelis]

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