Galois Theory, Second Edition

Series in Pure and Applied Mathematics
John Wiley & Sons

by

David A. Cox
Amherst College

What's in the Book

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. This undergraduate text develops the basic results of Galois theory, with Historical Notes to explain how the concepts evolved and Mathematical Notes to highlight the many ideas encountered in the study of this marvelous subject.

The book covers classic applications of Galois theory, such as solvability by radicals, geometric constructions, and finite fields. There are also 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. The book also explains how Maple and Mathematica can be used in computations related to Galois theory.

Later chapters explore the contributions of Lagrange, Galois, and Kronecker and describe how to compute Galois groups. There are also chapters on Galois's amazing results about irreducible polynomials of prime or prime-squared degree and Abel's wonderful theorem about geometric constructions on the lemniscate.

Typographical Errors in the First Edition

A list of typographical errors is available for Galois Theory: pdf or postscript .

Additional References to the First Edition

Here are some references to add to the chapter references in the first edition of the book:

• Section 7.4 discusses the inverse Galois problem and gives some references. An additional useful reference is the book Generic Polynomials: Constructive Aspects of the Inverse Galois Problem by C. U. Jensen, A. Ledet and N. Yui (Cambridge University Press, Cambridge, 2002).
• Section 9.1 discusses cyclotomic polynomials and mentions their coefficients in Example 9.1.7. The paper On the middle coefficient of a cyclotomic polynomial by G. P. Dresden (Amer. Math. Monthly 111 (2004), 531-533) discusses this topic and should be added to the references at the end of Section 9.1.
• Section 10.3 discusses origami constructions and lists some methods (such as marked rulers and intersections of conics) that are equivalent to origami. One equivalent construction not mentioned in the book involves mira. The paper Reflections on a mira by J. W. Emert, K. I. Meeks and R. B. Nelson (Amer. Math. Monthly 101 (1994), 544-549) discusses this topic and should be added to the references at the end of Section 10.3.
• Section 13.1 discusses how to compute the Galois group of a quartic polynomial, assuming that the field has characteristic different from 2. Keith Conrad has written a nice treatment of quartics (and cubics) that works for all characteristics. Click here to get a pdf copy of Keith's article.
• Section 13.2 discusses how to compute the Galois group of a quintic polynomial and in Example 13.2.13 mentions the problem of finding the roots of a quintic that is solvable by radicals. The paper Solving quintics by radicals by D. Lazard (in The Legacy of Niels Henrik Abel, O. Laudal and R. Piene, editors, Springer-Verlag, 2004, pp. 207-226) discusses methods for doing this and should be added to the references at the end of Section 13.2.

The Second Edition

For the second edition, the following changes have been made:

• Numerous typographical errors were corrected.
• Some exercises were dropped and others were added, a net gain of six.
• Section 13.3 contains a new subsection on the Galois group of irreducible separable quartics in all characteristics, based on the article of Keith Conrad mentioned above.
• The discussion of Maple in Section 2.3 was updated.
• Sixteen new references were added, including all of the additional references mentioned above.
• The notation section was expanded to include all notation used in the text.
• Appendix C on student projects was added at the end of the book.

Typographical Errors in the Second Edition