Starting with a fundamental introduction to the concepts and methods of algebra, the book presents the essential results and provides insight into many developments within algebra that are closely intertwined with other areas of mathematics. Beginning with concepts such as groups and rings, the book leads to field extensions and Galois theory. It then covers central parts of the theory of modules, algebras, and rings. The theory of division algebras and their classification using the Brauer group is developed. This is followed by introductions to algebraic number theory and the theory of quadratic forms. Numerous supplements offer perspectives on advanced topics. For example, general linear groups, skew polynomial rings, representations, extensions of modules, and projective spaces are discussed.