An introduction to the algebra of rings and fields

Abstract

This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and polynomials, the constructions of ring extensions and finite fields, some number-theoretical applications (such as a proof of quadratic reciprocity and Jacobsthal's formulas for $p=x^2+y^2$), and tastes of Gröbner bases and the Smith normal form. Familiarity with groups and vector spaces is assumed, though no deep results from either theory are used. Over 250 exercises are included (mostly without solutions).

arXiv comment583 pages. v2 adds sections 2.18 (regular elements), 2.19 (strong gcd-sequences, with a new take on factorization over divisors) and 6.6 (cyclotomic polynomials, with Little Wedderburn). Will grow further once I teach the course on a semester schedule. Comments are welcome!