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**Basic algebra I. 2nd ed.**
*(English)*
Zbl 0557.16001

New York: W. H. Freeman and Company. XVIII, 499 p. £19.95 (1985).

About ten years after publication of the first edition of this book (1974; Zbl 0284.16001), and five years after Part II (1980; Zbl 0441.16001), the author gives us a new edition of Part I. This book is well established, and only minor changes were necessary.

A glance at the table of contents shows a new paragraph on mod p reduction which facilitates the computation of the Galois group of an equation. In the same chapter ‘Galois theory’ the paragraph on finite fields is fully rewritten and gives now a proof (formerly in the exercises) of Gauß’ formula for the number of monic irreducible polynomials of degree n over a finite field.

‘Tarski’s principle’ has turned into ‘Tarski’s theorem’ (“Tariski” in the preface). A new, elementary, proof of the basic elimination theorem is also given in this chapter on formally real fields (a notion which is not mentioned before part II).

An appendix ‘Some topics for independent study’ is added. It contains references to the literature, e.g. about Mathieu groups, Hilbert’s irreducibility theorem, or Plücker equations. There are some new exercises. The excellent outlet of the book has not changed! (“Weddeburn” in the index also has not changed – but I think, on the whole, there are remarkable few misprints.)

It should be noted that a booklet ‘Solutions to selected exercises in Basic Algebra I’ prepared by Anthony G. Petrelle and the author refering to the first edition is available by the publishers.

A glance at the table of contents shows a new paragraph on mod p reduction which facilitates the computation of the Galois group of an equation. In the same chapter ‘Galois theory’ the paragraph on finite fields is fully rewritten and gives now a proof (formerly in the exercises) of Gauß’ formula for the number of monic irreducible polynomials of degree n over a finite field.

‘Tarski’s principle’ has turned into ‘Tarski’s theorem’ (“Tariski” in the preface). A new, elementary, proof of the basic elimination theorem is also given in this chapter on formally real fields (a notion which is not mentioned before part II).

An appendix ‘Some topics for independent study’ is added. It contains references to the literature, e.g. about Mathieu groups, Hilbert’s irreducibility theorem, or Plücker equations. There are some new exercises. The excellent outlet of the book has not changed! (“Weddeburn” in the index also has not changed – but I think, on the whole, there are remarkable few misprints.)

It should be noted that a booklet ‘Solutions to selected exercises in Basic Algebra I’ prepared by Anthony G. Petrelle and the author refering to the first edition is available by the publishers.

Reviewer: B.Richter

### MSC:

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

00A05 | Mathematics in general |

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |