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A concrete introduction to higher algebra. 3rd ed. (English) Zbl 1165.00002
Undergraduate Texts in Mathematics. New York, NY: Springer (ISBN 978-0-387-74527-5/hbk). xiv, 603 p. EUR 46.95/net; SFR 78.00; \$ 69.95; £ 37.99 (2009).

The content of this undergraduate textbook is as follows: 1. Numbers, 2. Induction, 3. Euclid’s Algorithm, 4. Unique Factorization, 5. Congruences, 6. Congruence Classes, 7. Rings and Fields, 8. Matrices and Codes, 9. Fermat’s and Euler’s Theorems, 10. Applications of Euler’s Theorem, 11. Groups, 12. The Chinese Remainder Theorem, 13. Polynomials, 14. Unique Factorization, 15. The Fundamental Theorem of Algebra, 16. Polynomials in $ℚ\left[x\right]$, 17. Congruences and the Chinese Remainder Theorem, 18. Fast Polynomial Multiplication, 19. Cyclic Groups and Cryptography, 20. Cardinal Numbers, 21. Quadratic Reciprocity, 22. Quadratic Application, 23. Congruence Classes Modulo a Polynomial, 24. Homomorphisms and Finite Fields, 25. BCH codes, 26. Factoring in $ℤ\left[x\right]$, 27. Irreducible polynomials. The book has a section with answers and hints to some exercises, a short reference list (about 100 items) and an index.

Prerequisites to follow this textbook are pre-calculus algebra and one year of calculus. The first six chapters are accessible to an average (European) high-schooler. The author says that the ultimate goal is to reach a substantial result in abstract algebra, namely, the classification of finite fields; this reviewer thinks that the crowning jewel of the book is the section on quadratic reciprocity law. The starting point is very elementary and passage to more complicated topics is fairly smooth. Every section is accompanied by a set of exercises, mostly easy, but some challenging. Motivating illustrations are often given by way of modern applications (within 50 years), notably almost all in computer domain (cryptography, etc). Alas the credit number check (Luhn’s algorithm) does not seem to work on any of this reviewer’s credit cards, perhaps indicating that the algorithm is no longer used for that purpose; Luhn’s patent application was in 1954 granted in 1960). An attempt is made to revisit same topics from different points of view, as the new material is developed. The author rightly teaches the reader that many deep theorems that are used today, and in applications at that, go back to the old Greece. The author made a wise choice to include actual formulas for roots of polynomial equations of 3rd and 4th degrees; so was also the choice of not-so-frequent-to-be-found estimates of polynomial roots in terms of their coefficients (but see more general estimates in: [R. Dimitrić, Math. Balk., New Ser. 11, No. 3–4, 203–206 (1997; Zbl 1032.12002)].

Reviewer’s remark: A spot-check revealed a few misprints. There are some problems common with (calculus) textbooks that have to cover a large stretch of the mathematical cultural territory within the short spectrum of abilities of an average (American) student. Thus the notions of range and codomain of functions are confused, as typically found in such textbooks.

“Determinants” go only to size 3 by 3. A pedagogical assumption is made throughout the book that more special is clearer, than more general. On p.172 a nonsensical claim is made that order of an element in a finite cyclic group and least common multiple of two numbers are “similar” notions...

##### MSC:
 00A05 General mathematics 11-01 Textbooks (number theory) 12-01 Textbooks (field theory) 94-01 Textbooks (information and communication)