This book aims to be a concise introduction to topics in commutative algebra, with an emphasis on worked examples and applications. It combines elegant algebraic theory with applications to number theory, problems in classical Greek geometry, and the theory of finite fields which has important uses in other branches of science. Topics covered include rings and Euclidean rings, the four-squares theorem, fields and field extensions, finite cyclic groups and finite fields. The material covered in this book prepares the way for the further study of abstract algebra, but it could also form the basis of an entire course.