Algebra is a subject we have become acquainted with during most of our mathematical education, often in connection with the solution of equations. Algebra: Groups, Rings, and Fields, Second Edition deals with developments related to their solutions.The principle at the heart of abstract algebra, a subject that enables one to deduce sweeping conclusions from elementary premises, is that the process of abstraction enables us to solve a variety of such problems with economy of effort. This leads to the glorious world of mathematical discovery.This second edition follows the original three-pronged approach: the theory of finite groups, number theory, and Galois' amazing theory of field extensions tying solvability of equations to group theory.As algebra has branched out in many directions, the authors strive to keep the text manageable while at the same time introducing the student to exciting new paths. In order to support this approach, the authors broadened the first edition, giving monoids a greater role, and relying more on matrices. Hundreds of new exercises were added.A course in abstract algebra, properly presented, could treat mathematics as an art as well as a science. In this exposition, we try to present underlying ideas, as well as the results they yield.