Description
Book SynopsisThis 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.
Trade Review“The book is very nicely written and the original style and choices of the topics make it agreeable reading, and might well complement and motivate the study of other classical introductions to the theory of more general number fields.” (Alessandro Cobbe, zbMATH 1498.11003, 2022)
Table of Contents1. 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.
