Rédei-Funktionen und das Schur'sche Problem. (Rédei functions and the Schur problem) (Q1115921)

The conjecture of Schur (which was fully proved by M. Fried in 1970) says that any polynomial g(x)\(\in {\mathbb{Z}}[x]\) which induces for infinitely many primes p a permutation of \({\mathbb{Z}}/(p)\) can be written as a composition of linear polynomials \(ax+b\in {\mathbb{Z}}[x]\), of powers \(x^ k\) and of Dickson polynomials \(g_ k(a,x)\in {\mathbb{Z}}[x].\) The author shows that there exists a class of rational functions over \({\mathbb{Z}}\), the so-called Rédei functions, which induce permutations of \({\mathbb{Z}}/(p)\) for infinitely many primes p. If n is an odd positive integer and \(\alpha\) an integer which is not a square in \({\mathbb{Z}}\) then two polynomials \(g_ n(x), h_ n(x)\in {\mathbb{Z}}[x]\) are determined by \((x+\sqrt{\alpha})^ n=g_ n(x)+h_ n(x)\sqrt{\alpha}\). The Rédei functions then are defined as \(f_ n=g_ n/h_ n\). The question to determine all rational functions over \({\mathbb{Z}}\) which induce permutations of \({\mathbb{Z}}/(p)\) for infinitely many p can be regarded as an extended Schur conjecture. The paper contributes to this problem.