This book is the second volume of a two-volume English edition of an algebra textbook originally published in Japanese in 2010–2011, and it develops more advanced topics beyond the foundations presented in the first volume. While the first volume covers basic subjects such as groups, rings, fields, and Galois theory, this volume explores broader and deeper areas of algebra that are essential for further study in algebraic geometry, number theory, and representation theory.

The main topics include transcendental extensions and infinite Galois extensions of fields, commutative algebra, completions of groups, rings and fields, finite-dimensional representations of finite groups, homological algebra, and simple rings and the Brauer group. These subjects are fundamental for understanding modern developments in algebra and its applications.

In addition to the main chapters, the appendices provide several advanced or specialized topics, such as the finite generation of invariant rings in characteristic 0, the construction of Witt rings and certain abelian extensions in positive characteristic, Gröbner bases, parametrization of field extensions by prehomogeneous vector spaces, and the distinction between symmetric bilinear forms and quadratic forms in characteristic 2. These topics, while relatively independent, offer valuable insights into important directions in contemporary algebra.

Throughout the book, the author emphasizes clarity through numerous examples and exercise problems, enabling readers to grasp abstract concepts effectively. This volume is intended for advanced undergraduate and graduate students seeking a deeper understanding of algebra, as well as for readers preparing to study algebraic geometry, number theory, and related fields.

Akihiko Yukie is a professor emeritus of Tohoku Univerisity and Kyoto University.