An introduction to the algebra of rings and fields

TL;DR

This paper provides an accessible introduction to rings and fields, covering fundamental concepts, constructions, applications, and advanced topics like Gr"obner bases, suitable for undergraduate students.

Contribution

It offers a comprehensive, undergraduate-level overview of algebraic structures with practical exercises and includes some advanced topics not typically covered in basic courses.

Findings

Includes proofs of quadratic reciprocity and formulas for sums of squares

Provides over 200 exercises for practice

Introduces advanced topics like Gr"obner bases and Smith normal form

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 Gr\"obner bases and the Smith normal form. Familiarity with groups and vector spaces is assumed, though no deep results from either theory are used. Over 200 exercises are included (mostly without solutions).