# Galois Theory, Second Edition

## 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.

## 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.
• 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.
The second edition was published in 2012.

## Typographical Errors in the Second Edition

Here are the typographical errors in the second edition known as of September 2018:
• Page 18: Two lines below the second display, "t3" should be "4t3".
• Page 38: On the second line of (2.22), "(-1)r-1" should be "(-1)n-1". [This error also appears in the first edition.]
• Page 88: Part (b) of Exercise 2 of Section 4.2 should assume that the leading coefficient of g(x) is not divisible by the prime p. [This error also appears in the first edition.]
• Page 113: On line 4, "deg(g)-1" should be "deg(gi)-1". [This error also appears in the first edition.]
• Page 132: In two places in Exercise 5, "Exercise 15 of Section 5.3" should be "Exercise 16 of Section 5.3".
• Page 201: On line 5 of Section 8.3, "Section 8.5" should be "Section 8.6"
• Page 428: Parts (b) and (c) of Exercise 7 of Section 14.2 need to be interchanged. Also, in the new part (c) (the former part (b)), the display should be followed with a new sentence "Here, (τ')-1 ⋅ φ is the action of A on H defined in part (b)." [This error also appears in the first edition.]
• Page 477: On line 1, "P2(u) = 1" should be "P2(u) = 2".
• Page 479: In the first display in the Historical Notes, the numerator in the formula for γ is incorrect. The "2" should be replaced with "2α". [This error also appears in the first edition.]
• Page 528: On the second line of Exercise 9, "n > 0" should be "n > 1". [This error also appears in the first edition.]
• Page 540: In the hint for Exercise 11 of Section 5.1, "root of F " should be "root of f ".
• Page 540: In the hint for Exercise 13 of Section 5.1, "Part (a) of Exercise 7" should be "Part (a) of Exercise 8".
• Page 566: In the index entry for permutation group, regular, "434, 439" should be "434, 439".

## Additional References to the Second Edition

Here are a reference to add to the chapter references in the second edition of the book:

• Chapter 14 discusses solvable permutation groups. The book Permutation Group Algorithms by Ákos Seress (Cambridge University Press, Cambridge, 2003) gives a graduate-level introduction to algorithms for dealing with permutation groups. Chapter 7 of this book focuses on the solvable case.
• In Chapter 15, the one-to-one group homomorphism constructed in Theorem 15.5.1 is actually an isomorphism. This is proved in the article The Galois theory of the lemniscate (J. Number Theory 135 (2014), 43-59) written with Trevor Hyde.

## Typographical Errors in the First Edition

A list of typographical errors is available for the first edition of Galois Theory: pdf or postscript . All of these errors were corrected in the second edition. You should also check the errata list for the second edition, since some these errors were present in the first edition.

## 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.
These references have been incorporated into the second edition.

## Ordering Information for the Second Edition

Hardcover Version. Click here for the Wiley catalog page for the second edition of Galois Theory. This page includes a brief description of the book and information on how to order a copy.

E-book Version. The second edition is also available as an e-book. Click here for the Wiley catalog page for the e-book version.

Individual Chapters. Chapters of the book can also be ordered individually -- Click here for the Wiley catalog page for individual chapters. The three appendices to the second edition are freely available at this site.

## Contacting the Author

You can contact the author at the following email address:

dacox@amherst.edu

The web site for the book is:

http://dacox.people.amherst.edu/galois.html