An introduction to one of the most celebrated theories of mathematics

Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox’s Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics.

Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel’s theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami. Anyone fascinated by abstract algebra will find careful discussions of such topics as:

• 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

With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike.

Cox (mathematics, Amherst College) covers both classic applications of the theory and some of the more novel approaches. He begins with polynomials in the theory's foundations, cubic equations, advancing to symmetric polynomials and the roots of polynomials. He proceeds explaining fields, including extension fields, normal and separable extensions, the Galois group, and Galois correspondence. Topics in applications include solvability by radicals, cyclotomic extensions, geometric constructions and finite fields. He then considers the work of Lagrange, Galois and Kronecker in concert, the process of computing Galois groups, solvable permutation groups, and the lemniscate, including the lemniscatic function, complex multiplication and Abel's theorem. He kindly provides information on abstract algebra as well as hints on selected exercises and a very well-organized bibliography.
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"This book provides a very detailed and comprehensive presentation of the theory and applications of Galois theory." (Mathematical Reviews, Issue 2006a)

"Happily, Cox's book reads more like a monograph, making a solid case for new subjects rather than rapidly treating a classical one." (CHOICE, September 2005)

" … offers a careful discussion … and will certainly fascinate anyone interested in abstract algebra: a remarkable book!" (Monatshefte fur Mathematik, August 2006)

DAVID A. COX is a professor of mathematics at Amherst College. He pursued his undergraduate studies at Rice University and earned his PhD from Princeton in 1975. The main focus of his research is algebraic geometry, though he also has interests in number theory and the history of mathematics. He is the author of Primes of the Form x2 + ny2, published by Wiley, as well as books on computational algebraic geometry and mirror symmetry.

Preface.

Notation.

PART I: POLYNOMIALS.

Chapter 1. Cubic Equations.

Chapter 2. Symmetric Polynomials.

Chapter 3. Roots of Polynomials.

PART II: FIELDS.

Chapter 4. Extension Fields.

Chapter 5. Normal and Separable Extensions.

Chapter 6. The Galois Group.

Chapter 7. The Galois Correspondence.

PART III: APPLICATIONS.

Chapter 8. Solvability by Radicals.

Chapter 9. Cyclotomic Extensions.

Chapter 10. Geometric Constructions.

Chapter 11. Finite Fields.

PART IV: FURTHER TOPICS.

Chapter 12. Lagrange, Galois, and Kronecker.

Chapter 13. Computing Galois Groups.

Chapter 14. Solvable Permutation Groups.

Chapter 15. The Lemniscate.

Appendix A: Abstract Algebra.

Appendix B: Hints to Selected Exercises.

References.

Index.