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

Jacobson, Nathan

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.

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

Keywords:

monoids; groups of transformations; Cayley theorem; permutations; orbits; generators; relations; Sylow theorems; matrix rings; quaternions; ideals; field of fractions; polynomial rings; polynomial functions; symmetric polynomials; principle ideal; domains; ring of endomorphisms; free modules; direct sums; finitely; generated modules; constructions with straight-edge and compass; splitting field; Galois group; Galois pairing; solvability by; radicals; transcendence of e and pi; finite fields; traces; norms; real closed fields; Sturm theorem; elimination; resultants; Tarski; theorem; bilinear forms; alternate forms; quadratic forms; Cartan-Dieudonné theorem; orthogonal groups; symplectic group; Hermitian forms; unitary geometry; determinants; Lie algebras; Jordan algebras; Hurwitz problem; Wedderburn theorem; division; algebras; distributive lattices; modular lattices; Jordan-Hölder-Dedekind theorem; Möbius function; number of monic irreducible polynomials

Citations:

Zbl 0284.16001; Zbl 0441.16001