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A Classical Introduction to Modern Number Theory

Kenneth Ireland and Michael Rosen
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics
BLL Rating: 

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

Ireland and Rosen's A Classical Introduction to Modern Number Theory hardly needs another good review from me... but hey, I'm going to give it one anyway. This is a great book, one that does exactly what it proposes to do, and does it well. For me, this is the go-to book whenever a student wants to do an advanced independent study project in number theory.

The title gives a good idea of what's in the book. The approach is "classical", which means fairly concrete and based on explicit, computable examples. But the topics are chosen with an eye to what is important in number theory today, from reciprocity laws to elliptic curves. The result is a book that does exactly what it proposes to do, opening up modern number theory to students who are not quite ready for group schemes and Galois representations.

In most universities and colleges, there is no course for which this book would be the ideal text. Courses on number theory tend to be either very elementary or too advanced and specialized. But for a student who wants to get started on the subject and has taken a basic course on elementary number theory and the standard abstract algebra course, this is perfect.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages. He somehow finds time to also be the editor of MAA Reviews.


1: Unique Factorization

2: Applications of Unique Factorization

3: Congruence

4: The Structure of U

5: Quadratic Reciprocity

6: Quadratic Gauss Sums

7: Finite Fields

8: Gauss and Jacobi Sums

9: Cubic and Biquadratic Reciprocity

10: Equations over Finite Fields

11: The Zeta Function

12: Algebraic Number Theory

13: Quadratic and Cyclotomic Fields

14: The Stickelberger Relation and the Eisenstein Reciprocity Law

15: Bernoulli Numbers

16: Dirichlet L-functions

17: Diophantine Equations

18: Elliptic Curves

19: The Mordell-Weil Theorem

20: New Progress in Arithmetic Geometry