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The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, diophantine dimensions of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.