Fermat, Class Field Theory, and Complex Multiplication

Amherst College

with contributions by Roger Lipsett

Paperback Edition, 1997

Second Edition, 2013

Third Edition with Solutions, 2022

The third edition of *Primes of the Form x ^{2} +
ny^{2}* was published in 2022 by the American Mathematical Society. The main features of the third edition are:

- Solutions to all exercises, written by Roger Lipsett and David Cox, are inlcuded.
- Section 15 on Shimura Reciprocity has been rewritten for clarity.
- There is now a unified Reference section.
- All exercises have been checked for clarity and correectness.
- A few new exercises and references have been added.
- All known errata in the second edition have been corrected.

The known typographical errors are:

- Page 50: On line –11, "one more
*n*(see Weinberger" should be "two more*n*(this follows from Weinberger" (this error also appears on page 56 of the second edition). - Page 98: On the first line of (6.9),
*p*_{i}* should be*p*_{1}* (this error also appears on page 112 of the second edition).

The second edition of *Primes of the Form x ^{2} +
ny^{2}* was published in 2013 by John Wiley & Sons. The main
features of the second edition are:

- There is a new Section 15 on Shimura reciprocity, based on the work of Peter Stevenhagen and Alice Gee.
- The new
*Additional References*section contains references not cited in the first edition. Some of these new references are mentioned in the text; the others are intended for further reading. - The new Exercise 12.31 at the end Section 12 shows how Ramanujam could have done one of Weber's singular moduli computations.
- All of the known errata in the first edition have been corrected.

The list of known typographical errors as of June 19, 2022 in the second edition is available in pdf .

Lists of typographical errors are available for the two printings
of the **hardback first edition**. To find out which printing you
have, check the bottom line on the copyright page; the last digit
displayed is the printing number.

- Typos for the first printing: pdf or postscript .
- Typos for the second printing: pdf or postscript .

For the **softcover first edition**, many (but not all) of the
typographical errors in the hardcover edition were corrected. The
remaining errors are available in
pdf or postscript .

All of these typos were corrected in the second edition of the book, which appeared in 2013.

For a nice set of exercises that supplement Chapter One of the book, click here. I am grateful to Jeffrey Stopple for preparing these exercises.

Click here
for the AMS catalog page for the third edition of **Primes of the Form x^{2} +
ny^{2}**. This page
includes a brief description of the
book and information on how to order a copy.

You can contact me at

dacox@amherst.edu