After a brief introduction reviewing the concepts of principal ideal domains and commutative fields, the book discusses residue classes (for example, the integers mog=dulo some number m); quadratic residues; algebraic integers (that is, objects that behave like integers in arbitrary algebraic structures), their discriminant; decomposition, norm, and classes of ideals; the ramification index; and the fundamental theorem of Abelian extensions. The theorems and definitions are carefully motivated, and the author frequently stops to explain how things fit together and what will come next. There are a great many exercises and many useful examples at a

From the reviews of the second edition:

"This book is a thorough self-contained introduction to algebraic number theory. The book is aimed at graduate students. The author made a great effort to make the subject easier to understand. The proofs are very detailed, there are plenty of examples and there are exercises at the end of almost all chapters . The book contains a great amount of material, more than enough for a year-long course." (Gábor Megyesi, Acta Scientiarum Mathematicarum, Vol. 69, 2003)

"There is a wealth of material in this book. The approach is very classical and global. the author keeps his presentation self-contained. The author has made a real effort to make the book accessible to students. Proofs are given in great detail, and there are many examples and exercises. The book would serve well as a text for a graduate course in classical algebraic number theory." (Lawrence Washington, Mathematical Reviews, Issue 2002 e)

"Ribenboims's 'Classical Theory of Algebraic Numbers' is an introduction to algebraic number theory on an elementary level . Ribenboim's book is a well written introduction to classical algebraic number theory and the perfect textbook for students who need lots of examples." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1082, 2006)

Autorentext
Paulo Ribenboim ist emeritierter Professor der kanadischen Queen's University, Fellow der Royal Society of Canada und Träger des George Pólya-Preises der Mathematical Association of America. Er ist Autor von 13 Büchern und über 150 Forschungsartikeln.

Inhalt
Unique Factorization Domains, Ideals, and Principal Ideal Domains.- Commutative Fields.- One.- Residue Classes.- Quadratic Residues.- Two.- Algebraic Integers.- Integral Basis, Discriminant.- The Decomposition of Ideals.- The Norm and Classes of Ideals.- Estimates for the Discriminant.- Units.- Extension of Ideals.- Algebraic Interlude.- The Relative Trace, Norm, Discriminant, and Different.- The Decomposition of Prime Ideals in Galois Extensions.- The Fundamental Theorem of Abelian Extensions.- Complements and Miscellaneous Numerical Examples.- Three.- Local Methods for Cyclotomic Fields.- Bernoulli Numbers.- Fermat's Last Theorem for Regular Prime Exponents.- More on Cyclotomic Extensions.- Four.- Characters and Gaussian Sums.- Zeta-Functions and L-Series.- The Dedekind Zeta-Function.- Primes in Arithmetic Progressions.- The Frobenius Automorphism and the Splitting of Prime Ideals.- Class Numbers of Quadratic Fields.- Class Number of Cyclotomic Fields.- Miscellaneous Results About the Class Number of Quadratic Fields.- Miscellaneous Results About the Class Number of Cyclotomic Fields.