Algebra

The central theme of the book is the study of algebraic equations and their corresponding field extensions, enriched by some more advanced topics that are optional. In particular, we discuss the basics on finite and algebraic extensions, on separable and purely inseparable extensions, as well as on splitting fields and the construction of algebraic closures. On a second stage, we give a comprehensive account on Galois theory, including the key techniques necessary for characterizing the solvability of algebraic equations via their corresponding Galois groups. There are some further applications of Galois theory, like a proof of the Fundamental Theorem of Algebra, a discussion of compass and straightedge constructions, as well as the explicit solution formulas for algebraic equations of degrees 3 and 4 from the 16 th century. Furthermore, we have included a historical introduction to the problem of solving algebraic equations at the very beginning of the book. Whenever appropriate, we have enriched chapters by additional material that leads to a deeper understanding of the subject. Such sections are indicated by a star ; they are more demanding and can be skipped on first reading. For example, this way we discuss the theory of elementary divisors, integral ring extensions, some beginnings of Algebraic Geometry, profinite Galois groups, symmetric polynomials including discriminant and resultant, multiplicative and general Kummer theory including Witt vectors, Galois descent, as well as, at the end of the book, a chapter on transcendental extensions. The whole text is self-contained, up to a few rudimentary facts from Linear Algebra. In particular, we have gathered in separate chapters all necessary prerequisites on groups, as well as on rings and polynomials. Each chapter of the book starts with a motivating introduction entitled "Background and Overview", which contains an informal discussion of the subjects dealt with in subsequent sections. All sections end with a list of specially adapted exercises, some of them with solutions in the Appendix. The book is an English translation and revision of the eighth edition of my German book on Algebra. The latter appeared for the first time in 1993 and, in later years, was modified and polished substantially in order to present things in a way as simple as possible, however, without resorting to simplifying and inappropriate ad hoc solutions.