Algebra

- Foreword

- Constructibility with Ruler and Compass

- Algebraic Extensions

- Simple Extensions

- Fundamentals of Divisibility

- Prime Factorization in Polynomial Rings. Gauss's Theorem

- Polynomial Splittin 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

- The Transcendence of pi

- Transcendental Field Extensions

- Hilbert's Nullstellensatz

- Appendix: Problems and Remarks

- Index of Notation

- Index.

From Math Reviews:
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 book ends with an appendix containing exercises and notes on the
previous parts of the book. However, brief historical comments and
suggestions for further reading are also scattered through the text.

From the reviews:

"This is the faithful English translation of the German textbook a ~EinfA1/4hrung in die Algebra: Teil Ia (TM) by F. Lorenz a ] . this introduction to classical abstract algebra quickly became one of the most popular texts a ] . this outstanding textbook has been brought about by Springer Verlag, with some further improvements and in modern printing. a ] the author offers quite a number of interesting methodological approaches, extra topics, and arithmetical applications. a ] The present English edition is enhanced by a carefully compiled index of notation." (Werner Kleinert, Zentralblatt MATH, Vol. 1087, 2006)

"This textbook is a translation of the 1987 German edition. It is an introduction into the classical parts of algebra with a focus on fields and Galois theory. The clear and well written exposition requires only some basics from linear algebra. a ] The book ends with an appendix containing exercises and notes on the previous parts of the book. It also includes brief historical comments. It can be highly recommended." (A. Winterhof, Internationale Mathematische Nachrichten, Issue 206, 2007)

"Algebra I: Fields and Galois Theory is a ] a complete algebra course, including both undergraduate and graduate topics. a ] uses Galois theory as a guiding line that leads the reader to excursions into arithmetic groups, rings, and of course fields. a ] the choice of topics and their organization are excellent and provide a unifying view of most of algebra. In all, Lorenza (TM)s book is a wonderful reference for both teachers and researches, and can be used with much profit for independent study by hard-working students." (Luiz Hendrique de Figueiredo, MathDL, July, 2008)