The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear 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.

Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students. From Reviews of the German version: This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. - Stefan Porubsky, Mathematical Reviews

Klappentext

This translation of the 1987 German edition is an introduction into the classical parts of algebra with a focus on fields and Galois theory. It discusses nonstandard topics, such as the transcendence of pi, and new concepts are defined in the framework of the development of carefully selected problems. It includes an appendix with exercises and notes on the previous parts of the book, and brief historical comments are scattered throughout.

Inhalt

Constructibility with Ruler and Compass.- Algebraic Extensions.- Simple Extensions.- Fundamentals of Divisibility.- Prime Factorization in Polynomial Rings. Gauss's Theorem.- Polynomial Splitting Fields.- Separable Extensions.- Galois Extensions.- Finite Fields, Cyclic Groups and Roots of Unity.- Group Actions.- Applications of Galois Theory to Cyclotomic Fields.- Further Steps into Galois Theory.- Norm and Trace.- Binomial Equations.- Solvability of Equations.- Integral Ring Extensions with Applications to Galois Theory.- The Transcendence of ?.- Fundamentals of Transcendental Field Extensions.- Hilbert's Nullstellensatz.