##
**A survey of modern algebra. 4th ed.**
*(English)*
Zbl 0365.00006

New York: Macmillan Publishing Co., Inc. London: Collier Macmillan Publishers. xi, 500 p. £10.15 (1977).

It is a pleasure to welcome a new edition of this standard text. It scarcely differs from the third edition published in 1965 [second edition (1953; Zbl 0052.25402)], any changes being of a minor nature. Indeed, this new edition could be viewed merely as a slightly polished version of the earlier one.

Some few parts of the text have been rewritten. For example, the section entitled “Some basic concepts of logic” is now called “Sets, functions and relations” and the first part of the section has been rewritten with the word “class” being dropped and functions being introduced. Also throughout “many-one correspondences” are now called functions or mappings. Some of the material has been rearranged. For example, the section on quotient spaces of vector spaces which was in chapter 8 now appears in chapter 7 after the discussion of normal orthogonal bases. And bilinear functions and tensor products have been moved from chapter 7 to chapter 8 after equivalence and canonical forms. Also chapter 15 (Galois theory) has been slightly rearranged. Most of the text, however, is identical with its predecessor. Even the exercises are virtually unchanged with some dropping out and a few new ones being introduced, less than ten in each case.

The new edition is nicely produced and the final appearance is very pleasing. The text now occupies 500 pages compared to 437 pages in the third edition and there is a general feeling that the material is not as crammed together as it was before. Some minor errors have been corrected (e.g. exercise 9 of §2.1 is now correct) but others have been perpetrated (e.g. “trangles” for “triangles” in exercise 9 of §1.8).

The authors might be criticised for not taking this opportunity to revise the text more drastically since Algebra has not been static in the thirteen years since the previous edition. For example, rings with polynomial identity are not mentioned. Yet who can blame them for sticking with a successful formula?

This is still a nice textbook of elementary Algebra for those who don’t have the third (or earlier) edition but, if you do, the changes scarcely warrant its purchase which is a relief in view of the price!

Some few parts of the text have been rewritten. For example, the section entitled “Some basic concepts of logic” is now called “Sets, functions and relations” and the first part of the section has been rewritten with the word “class” being dropped and functions being introduced. Also throughout “many-one correspondences” are now called functions or mappings. Some of the material has been rearranged. For example, the section on quotient spaces of vector spaces which was in chapter 8 now appears in chapter 7 after the discussion of normal orthogonal bases. And bilinear functions and tensor products have been moved from chapter 7 to chapter 8 after equivalence and canonical forms. Also chapter 15 (Galois theory) has been slightly rearranged. Most of the text, however, is identical with its predecessor. Even the exercises are virtually unchanged with some dropping out and a few new ones being introduced, less than ten in each case.

The new edition is nicely produced and the final appearance is very pleasing. The text now occupies 500 pages compared to 437 pages in the third edition and there is a general feeling that the material is not as crammed together as it was before. Some minor errors have been corrected (e.g. exercise 9 of §2.1 is now correct) but others have been perpetrated (e.g. “trangles” for “triangles” in exercise 9 of §1.8).

The authors might be criticised for not taking this opportunity to revise the text more drastically since Algebra has not been static in the thirteen years since the previous edition. For example, rings with polynomial identity are not mentioned. Yet who can blame them for sticking with a successful formula?

This is still a nice textbook of elementary Algebra for those who don’t have the third (or earlier) edition but, if you do, the changes scarcely warrant its purchase which is a relief in view of the price!

Reviewer: Patrick F. Smith (Glasgow)

### MSC:

00A05 | Mathematics in general |

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

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

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

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

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