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Dickson polynomials. (English) Zbl 0823.11070

Pitman Monographs and Surveys in Pure and Applied Mathematics. 65. Harlow: Longman Scientific & Technical. New York, NY: John Wiley & Sons, Inc.. 207 p. (1993).
L. E. Dickson’s Ph.D. thesis in 1896 contained information on classes of polynomials of the form \[ F(x,n)= x^ n+n \sum_{i=1}^{(n -1)/2} \Biggl( {{(n-i-1) \ldots (n- 2i+1)} \over {i!}} \Biggr) a^ i x^{n- 2i} \] over finite fields for odd \(n\in \mathbb{Z}\). Some of the information was a partial proof that \(F(x,n)\) induces a permutation of \(\mathbb{F}_ q\) (the field of order \(q\)), if and only if \(\text{gcd} (n, q^ 2-1) =1\). This began a study of such polynomials, which I. Schur has since dubbed Dickson polynomials (D-p’s), and have been the object of study for a century now.
This book is dedicated to putting, in one location, all results concerning D-p’s, and their applications. The authors intend it to be a research monograph, although they do provide relevant background in Chapter 1, so that the book is self-contained. They devote Chapter 2 to the basic properties of D-p’s, and Chapter 3 deals with these polynomials over finite fields. Properties of polynomials over residue class rings of \(\mathbb{Z}/ (m)\) is the focus of Chapter 4, whereas Chapter 5 deals with more general rings.
In 1923, Schur published a paper which gave rise to the conjecture that the only polynomials with integer coefficients which induce permutations of \(\mathbb{Z}/ (p)\) for infinitely many primes \(p\), are compositions of linear polynomials, power polynomials, and D-p’s. Schur’s conjecture was proved in 1970 by M. Fried. Chapter 6 is devoted to giving the first elementary proof of Schur’s conjecture. Applications of D-p’s to cryptology and primality testing are given in Chapter 7.
Exercises are provided as well as a collection of notes at the end of each chapter.
The book is certainly an excellent reference text for anyone interested in D-p’s. However, it does fail to contain some relatively recent research on the topic such as the modest contribution by the reviewer and C. Small [Int. J. Math. Math. Sci. 10, 535-544 (1987; Zbl 0626.12015)].
Reviewer: R.Mollin (Calgary)

MSC:

11T06 Polynomials over finite fields
12E05 Polynomials in general fields (irreducibility, etc.)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
13B25 Polynomials over commutative rings
11Txx Finite fields and commutative rings (number-theoretic aspects)

Citations:

Zbl 0626.12015
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