Algebra I

This book is the first volume of a two-volume English edition of an algebra textbook originally published in Japanese in 2010-2011. It begins with a review of set theory and develops the core subjects of undergraduate algebra, including group theory, ring theory, modules over rings, field theory, and Galois theory. The volume aims not only to cover fundamental material but also to guide readers toward significant goals such as the proof of the unsolvability of the general quintic equation by radicals.

In addition to the theory of equations, the book emphasizes the introduction and clarification of the basic notions of algebra. One of the main challenges in learning algebra lies not in computational difficulty but in understanding abstract concepts such as sets and logical reasoning. To address this, the first chapter revisits set theory with carefully chosen examples, including instructive incorrect solutions encountered in undergraduate teaching, which help illuminate common misunderstandings.

Maintaining motivation is another difficulty, particularly in the study of group theory. To overcome this, the book explains the broader significance of group theory and illustrates how it connects to Galois theory and more advanced topics such as the construction of moduli. While this volume focuses on foundational undergraduate material, it also prepares the reader for the more advanced topics treated in the second volume, which is intended for graduate-level study and includes connections to algebraic geometry and number theory.