Quadratic Number Fields, 1st ed. 2021 Springer Undergraduate Mathematics Series

This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level.

Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study.

Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.

1. Prehistory.- 2 Quadratic Number Fields.- 3 The Modularity Theorem.- 4 Divisibility in Integral Domains.- 5 Arithmetic in some Quadratic Number Fields.- 6 Ideals in Quadratic Number Fields.- 7 The Pell Equation.- 8 Catalan's Equation.- 9 Ambiguous Ideal Classes and Quadratic Reciprocity.- 10 Quadratic Gauss Sums.- A Computing with Pari and Sage.- B Solutions.- Bibliography.- Name Index.- Subject Index.

Franz Lemmermeyer has worked in algebraic number theory and has published several books on the history of number theory, in particular on reciprocity laws.

Connects quadratic fields with modern algebraic number theory Applies the theory to solve Diophantine equations Contains hundreds of exercises with solutions Includes original historical commentary