On a Problem concerning Permutation Polynomials

Let S(f) denote the set of integral ideals / such that / is a permutation polynomial modulo i", where / is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets S which may be written in the form S(f). Introduction. A polynomial f(x) with coefficients in a commutative ring R is said to be a permutation polynomial modulo an ideal I oi R (abbreviated p.p. mod /) if the mapping induced on the residue class ring R/I is bijective. From now on we assume that R is the ring of algebraic integers in an algebraic number field K (of finite degree). Put S\{f) = {P | P is a nonzero prime ideal such that f(x) is a p.p. modP but not modP2}, Si(f) = {P | P is a nonzero prime ideal and f(x) is a p.p. modP2}. Then f(x) is a p.p. modi (^ {0}) if and only if every prime divisor of / belongs to Si(/) U S2(/) and I is not divisible by the square of an element of Si(f) (cf. Lemma 1.1). It is the purpose of this paper to describe the sets Si, S2 that may be written in the form Si(/), 82(f) for some polynomial f(x) Taking R to be the ring of rational integers yields the solution of problem II posed by Narkiewicz in [4, p. 13]. Denoting the absolute norm of an ideal / by NI (= \R/I\), we obtain the following characterization: THEOREM. Let R be the ring of integers in the algebraic number field K. If Si,S2 are disjoint sets of nonzero prime ideals of R then there exists a polynomial f(x) e R[x] such that St = Si(f) (i — 1,2) if and only if one of the following conditions holds: (1) Si,52 are finite. (2) For some squarefree positive integer n with (n, 6) = 1 we have Si is a finite set of prime ideals P such that NP ^ l(n) or 2TM_1 = 0 (P); S2 differs from {P \ (NP2 — l,n) = 1} by at most finitely many elements. (3) For some positive integers m,n with (n, 6) = (m, 2) = 1, mn > 1, mn squarefree, we have Si differs from {P | (NP — l,m) = (NP2 — 1, n) = 1} by at most finitely many prime ideals P with NP ^ l(mn) or 2"_1 = 0 (P); S2 is finite. (Note that 2""1 = 0 (P) is equivalent to n > 1 and 2 = 0(P).) The theorem is an immediate consequence of Proposition 2.2, Proposition 2.13, and Proposition 4.8. For the "only if" part we make use of Fried's proof of Schur's conjecture. In §3 Received by the editors June 26, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11T06; Secondary 11R09.