*Algebraic number theory* involves using techniques from (mostly
commutative) algebra and finite group theory to gain a deeper
understanding of the arithmetic of number fields and related objects
(e.g., functions fields, elliptic curves, etc.). The main objects that we
study in this book are number fields, rings of integers
of number fields, unit groups, ideal class groups, norms, traces,
discriminants, prime ideals, Hilbert and other class fields and
associated reciprocity laws, zeta and -functions, and algorithms
for computing each of the above.

William Stein 2012-09-24