Galois Theory 5th Edition by Ian Stewart 1032101598 9781032101590

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ISBN 10: 1032101598

ISBN 13: 9781032101590

Author: Ian Stewart

Since 1973, Galois theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students. New to the Fifth Edition Reorganised and revised Chapters 7 and 13 New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations This bestseller continues to deliver a rigorous, yet engaging, treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.

Galois Theory 5th Table of contents:

1 Classical Algebra

1.1 Complex Numbers

1.2 Subfields and Subrings of the Complex Numbers

1.3 Solving Equations

1.4 Solution by Radicals

2 The Fundamental Theorem of Algebra

2.1 Polynomials

2.2 Fundamental Theorem of Algebra

2.3 Implications

3 Factorisation of Polynomials

3.1 The Euclidean Algorithm

3.2 Irreducibility

3.3 Gauss’s Lemma

3.4 Eisenstein’s Criterion

3.5 Reduction Modulo p

3.6 Zeros of Polynomials

4 Field Extensions

4.1 Field Extensions

4.2 Rational Expressions

4.3 Simple Extensions

5 Simple Extensions

5.1 Algebraic and Transcendental Extensions

5.2 The Minimal Polynomial

5.3 Simple Algebraic Extensions

5.4 Classifying Simple Extensions

6 The Degree of an Extension

6.1 Definition of the Degree

6.2 The Tower Law

6.3 Primitive Element Theorem

7 Ruler-and-Compass Constructions

7.1 Approximate Constructions and More General Instruments

7.2 Constructions in ℂ

7.3 Specific Constructions

7.4 Impossibility Proofs

7.5 Construction from a Given Set of Points

8 The Idea behind Galois Theory

8.1 A First Look at Galois Theory

8.2 Galois Groups According to Galois

8.3 How to Use the Galois Group

8.4 The Abstract Setting

8.5 Polynomials and Extensions

8.6 The Galois Correspondence

8.7 Diet Galois

8.8 Natural Irrationalities

9 Normality and Separability

9.1 Splitting Fields

9.2 Normality

9.3 Separability

10 Counting Principles

10.1 Linear Independence of Monomorphisms

11 Field Automorphisms

11.1 K-Monomorphisms

11.2 Normal Closures

12 The Galois Correspondence

12.1 The Fundamental Theorem of Galois Theory

13 Worked Examples

13.1 Examples of Galois Groups

13.2 Discussion

14 Solubility and Simplicity

14.1 Soluble Groups

14.2 Simple Groups

14.3 Cauchy’s Theorem

15 Solution by Radicals

15.1 Radical Extensions

15.2 An Insoluble Quintic

15.3 Other Methods

16 Abstract Rings and Fields

16.1 Rings and Fields

16.2 General Properties of Rings and Fields

16.3 Polynomials Over General Rings

16.4 The Characteristic of a Field

16.5 Integral Domains

17 Abstract Field Extensions and Galois Groups

17.1 Minimal Polynomials

17.2 Simple Algebraic Extensions

17.3 Splitting Fields

17.4 Normality

17.5 Separability

17.6 Galois Theory for Abstract Fields

17.7 Conjugates and Minimal Polynomials

17.8 The Primitive Element Theorem

17.9 Algebraic Closure of a Field

18 The General Polynomial Equation

18.1 Transcendence Degree

18.2 Elementary Symmetric Polynomials

18.3 The General Polynomial

18.4 Cyclic Extensions

18.5 Solving Equations of Degree Four or Less

18.6 Explicit Formulas

19 Finite Fields

19.1 Structure of Finite Fields

19.2 The Multiplicative Group

19.3 Counterexample to the Primitive Element Theorem

19.4 Application to Solitaire

20 Regular Polygons

20.1 What Euclid Knew

20.2 Which Constructions are Possible?

20.3 Regular Polygons

20.4 Fermat Numbers

20.5 How to Construct a Regular 17-gon

21 Circle Division

21.1 Genuine Radicals

21.2 Fifth Roots Revisited

21.3 Vandermonde Revisited

21.4 The General Case

21.5 Cyclotomic Polynomials

21.6 Galois Group of ℚ(ζ)/ℚ

21.7 Constructions Using a Trisector

22 Calculating Galois Groups

22.1 Transitive Subgroups

22.2 Bare Hands on the Cubic

22.3 The Discriminant

22.4 General Algorithm for the Galois Group

23 Algebraically Closed Fields

23.1 Ordered Fields and Their Extensions

23.2 Sylow’s Theorem

23.3 The Algebraic Proof

24 Transcendental Numbers

24.1 Irrationality

24.2 Transcendence of e

24.3 Transcendence of π

25 What Did Galois Do or Know?

25.1 List of the Relevant Material

25.2 The First Memoir

25.3 What Galois Proved

25.4 What is Galois up to?

25.5 Alternating Groups, Especially A5

25.6 Simple Groups Known to Galois

25.7 Speculations about Proofs

25.8 A5 is Unique

26 Further Directions

26.1 Inverse Galois Problem

26.2 Differential Galois Theory

26.3 p-adic Numbers

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