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
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