Modern algebra. Vols. I, II.

*(English)*Zbl 0134.24903
Prentice-Hall Mathematics Series. Englewood Cliffs, N.J.: Prentice-Hall, Inc. x, 806 p. (1965).

This book is intended primarily as an introduction to abstract algebra for undergraduates. The six chapters of Vol. I contain the usual material for such introductory texts, while the five chapters of Vol. II which hardly depend on each other are of a more specialized nature. Each chapter is divided into sections followed by exercises giving in all an impressive total of over 1300 exercises of varying degrees of difficulty. The author and publishers are to be congratulated on producing a splendid text.

The chapter headings for Vol. I are Algebraic structures, New structures from old, The natural numbers, Rings and fields, Vector spaces, Polynomials.

Chapter VII (The real and complex number fields) is concerned with ordered fields. \(\mathbb R\) is obtained from \(\mathbb Q\) by means of Cauchy sequences, and the chapter closes by showing that the only finite division algebras over \(\mathbb R\) are \(\mathbb R\), \(\mathbb C\) and the quaternions together with the generalization to real-closed totally ordered fields. Chapter VIII (Algebraic extensions of fields) is largely Galois theory, and concludes with an example of a quintic not solvable by radicals.

Chapter IX (Linear operators) contains the Primary and Rational Decomposition Theorems for linear operators on a finite dimensional vector space, together with a section on determinants which closes with the Cayley-Hamilton-Theorem. Chapter X (I) includes the Gram-Schmidt orthogonalization process, the Spectral Theorem and the Polar Decomposition Theorem for linear operators. Chapter XI (The Axiom of Choice) establishes the equivalence of the Axiom of Choice, the Hausdorff Maximality Principle, Zorn’s Lemma and Zermelo’s Theorem. It is shown that Zorn’s Lemma implies that every vector space has a basis, every formally real field has a unique ordering and every field has a unique algebraic closure.

The chapter headings for Vol. I are Algebraic structures, New structures from old, The natural numbers, Rings and fields, Vector spaces, Polynomials.

Chapter VII (The real and complex number fields) is concerned with ordered fields. \(\mathbb R\) is obtained from \(\mathbb Q\) by means of Cauchy sequences, and the chapter closes by showing that the only finite division algebras over \(\mathbb R\) are \(\mathbb R\), \(\mathbb C\) and the quaternions together with the generalization to real-closed totally ordered fields. Chapter VIII (Algebraic extensions of fields) is largely Galois theory, and concludes with an example of a quintic not solvable by radicals.

Chapter IX (Linear operators) contains the Primary and Rational Decomposition Theorems for linear operators on a finite dimensional vector space, together with a section on determinants which closes with the Cayley-Hamilton-Theorem. Chapter X (I) includes the Gram-Schmidt orthogonalization process, the Spectral Theorem and the Polar Decomposition Theorem for linear operators. Chapter XI (The Axiom of Choice) establishes the equivalence of the Axiom of Choice, the Hausdorff Maximality Principle, Zorn’s Lemma and Zermelo’s Theorem. It is shown that Zorn’s Lemma implies that every vector space has a basis, every formally real field has a unique ordering and every field has a unique algebraic closure.

Reviewer: David Kirby (Southampton)

##### MSC:

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

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

00A05 | Mathematics in general |