Praise for the First Edition

". . .will certainly fascinate anyone interested in abstract
algebra: a remarkable book!"

--Monatshefte fur Mathematik

Galois theory is one of the most established topics in
mathematics, with historical roots that led to the development of
many central concepts in modern algebra, including groups and
fields. Covering classic applications of the theory, such as
solvability by radicals, geometric constructions, and finite
fields, Galois Theory, Second Edition delves into novel
topics like Abel's theory of Abelian equations, casus
irreducibili, and the Galois theory of origami.

In addition, this book features detailed treatments of several
topics not covered in standard texts on Galois theory,
including:

* The contributions of Lagrange, Galois, and Kronecker

* How to compute Galois groups

* Galois's results about irreducible polynomials of prime
or prime-squared degree

* Abel's theorem about geometric constructions on the
lemniscates

* Galois groups of quartic polynomials in all
characteristics

Throughout the book, intriguing Mathematical Notes and
Historical Notes sections clarify the discussed ideas and
the historical context; numerous exercises and examples use Maple
and Mathematica to showcase the computations related to Galois
theory; and extensive references have been added to provide readers
with additional resources for further study.

Galois Theory, Second Edition is an excellent book for
courses on abstract algebra at the upper-undergraduate and graduate
levels. The book also serves as an interesting reference for anyone
with a general interest in Galois theory and its contributions to
the field of mathematics.

DAVID A. COX, PhD, is Professor in the Department of Mathematics at Amherst College. He has published extensively in his areas of research interest, which include algebraic geometry, number theory, and the history of mathematics. Dr. Cox is consulting editor for Wiley's Pure and Applied Mathematics book series and the author of Primes of the Form x2 + ny2 (Wiley).

Praise for the First Edition

". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!"
Monatshefte fur Mathematik

Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel's theory of Abelian equations, casus irreducibili, and the Galois theory of origami.

In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including:

• The contributions of Lagrange, Galois, and Kronecker

• How to compute Galois groups

• Galois's results about irreducible polynomials of prime or prime-squared degree

• Abel's theorem about geometric constructions on the lemniscate

• Galois groups of quartic polynomials in all characteristics

Throughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study.

Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.

". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!"
Monatshefte fur Mathematik

Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel's theory of Abelian equations, casus irreducibili, and the Galois theory of origami.

In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including:

• The contributions of Lagrange, Galois, and Kronecker
• How to compute Galois groups
• Galois's results about irreducible polynomials of prime or prime-squared degree
• Abel's theorem about geometric constructions on the lemniscates
• Galois groups of quartic polynomials in all characteristics

Throughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study.

Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.

Preface to the First Edition xvii

Preface to the Second Edition xxi

Notation xxiii

1 Basic Notation xxiii

2 Chapter-by-Chapter Notation xxv

PART I POLYNOMIALS

1 Cubic Equations 3

1.1 Cardan's Formulas 4

1.2 Permutations of the Roots 10

1.3 Cubic Equations over the Real Numbers 15

2 Symmetric Polynomials 25

2.1 Polynomials of Several Variables 25

2.2 Symmetric Polynomials 30

2.3 Computing with Symmetric Polynomials (Optional) 42

2.4 The Discriminant 46

3 Roots of Polynomials 55

3.1 The Existence of Roots 55

3.2 The Fundamental Theorem of Algebra 62

PART II FIELDS

4 Extension Fields 73

4.1 Elements of Extension Fields 73

4.2 Irreducible Polynomials 81

4.3 The Degree of an Extension 89

4.4 Algebraic Extensions 95

5 Normal and Separable Extensions 101

5.1 Splitting Fields 101

5.2 Normal Extensions 107

5.3 Separable Extensions 109

5.4 Theorem of the Primitive Element 119

6 The Galois Group 125

6.1 Definition of the Galois Group 125

6.2 Galois Groups of Splitting Fields 130

6.3 Permutations of the Roots 132

6.4 Examples of Galois Groups 136

6.5 Abelian Equations (Optional) 143

7 The Galois Correspondence 147

7.1 Galois Extensions 147

7.2 Normal Subgroups and Normal Extensions 154

7.3 The Fundamental Theorem of Galois Theory 161

7.4 First Applications 167

7.5 Automorphisms and Geometry (Optional) 173

PART III APPLICATIONS

8 Solvability by Radicals 191

8.1 Solvable Groups 191

8.2 Radical and Solvable Extensions 196

8.3 Solvable Extensions and Solvable Groups 201

8.4 Simple Groups 210

8.5 Solving Polynomials by Radicals 215

8.6 The Casus Irreducb…