Galois Theory, Second Edition
Series in Pure and Applied Mathematics
John Wiley & Sons
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
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:
The second edition was published in 2012.
- Numerous typographical errors were corrected.
- Some exercises were dropped and others were added, a net gain of
- 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.
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
- 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
- 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
- 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
- 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
- Page 528: On the second line of Exercise 9, "n > 0" should
be "n > 1". [This error also appears in the
- 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
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:
These references have been incorporated into the second edition.
- 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,
- 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.
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
for the Wiley catalog page for the e-book version.
Chapters of the book can also be ordered individually -- Click
for the Wiley catalog page for individual chapters. The
three appendices to the second edition are freely available at this site.
You can contact the author at the following email address:
The web site for the book is: