Theory of Algebraic Numbers

Detailed proofs and clear-cut explanations provide an excellent introduction to the elementary components of classical algebraic number theory in this concise, well-written volume.The authors, a pair of noted mathematicians, start with a discussion of divisibility and proceed to examine Gaussian primes (their determination and role in Fermat's theorem), polynomials over a field (including the Eisenstein irreducibility criterion), algebraic number fields, bases (finite extensions, conjugates and discriminants, and the cyclotomic field), and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields, the fundamental theorem of ideal theory and its consequences, ideal classes and class numbers, and the Fermat conjecture (concluding with discussions of Pythagorean triples, units in cyclotomic fields, and Kummer's theorem). In addition to a helpful list of symbols and an index, a set of carefully chosen problems appears at the end of each chapter to reinforce mathematics covered. Students and teachers of undergraduate mathematics courses will find this volume a first-rate introduction to algebraic number theory.