Algebraic Theory of Quadratic Numbers

This book shows the techniques of elementary arithmetic, ring theory and linear algebra working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. Includes numerous exercises.

By focusing on quadratic numbers, this advanced undergraduate or master's level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes.

The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory.

Offers an accessible introduction to number theory by focusing on quadratic numbers Includes many exercises that provide students with hands-on computational experience with quadratic number fields Presents a modern treatment of binary quadratic forms Includes supplementary material: sn.pub/extras

Autorentext

Mak Trifkovi is an assistant professor of mathematics at the University of Victoria.

Inhalt
1 Examples.- 2 A Crash Course in Ring Theory.- 3 Lattices.- 4 Arithmetic in Q[***D*].- 5 The Ideal Class Group and Geometry of Numbers.- 6 Continued Fractions.- 7 Quadratic Forms.- Appendix.- Hints to Selected Exercises.- Index.