Permutation polynomials over algebraic number fields
type: Item
instance of: scholarly article
title: Permutation polynomials over algebraic number fields
zbMATH Open document ID: 0533.12003
DOI: 10.1016/0022-314X(84)90060-X
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: If K is an algebraic number field with ring of integers A, then \textit{M. Fried} [Mich. Math. J. 17, 41-55 (1970; Zbl 0169.377)] has shown that any polynomial over A having the Schur property, i.e. inducing a permutation on infinitely many residue class fields of A, must be a composition of special polynomials, namely of cyclic polynomials \(\alpha x^ s+\beta\) with \(\alpha\),\(\beta\in A\), \(\alpha\neq 0\), and Chebyshev polynomials \(g_ t(x)\). The reviewer and \textit{S. K. Lo} [Abh. Math. Semin. Univ. Hamb. 49, 126-139 (1979; Zbl 0426.12001)] characterized the cyclic polynomials and Chebyshev polynomials having the Schur property in the cases of quadratic fields and cyclotomic fields K, and also in some other special cases. The author shows first that it suffices to study the Schur property for compositions of the form \(x^ s{\mathbb{O}}g_ t(x)\). A polynomial of this form not satisfying the Schur property is called a finite Schur polynomial for K. A finite Schur polynomial \(x^ s{\mathbb{O}}g_ t(x)\) for K is called a primitive Schur polynomial for K if there is no pair (s',t') with s'\(| s,t'| t\), \(s't'<st\), such that \(x^{s'}{\mathbb{O}}g_{t'}(x)\) is a finite Schur polynomial for K. (We have corrected two awkward misprints in this definition.) The following result provides a reduction to Abelian number fields: the finite Schur polynomials for K are exactly the finite Schur polynomials for the maximal Abelian subfield of K. If \(\gcd(2,s)=\gcd(6,t)=1\) and \(x^ s{\mathbb{O}}g_ t(x)\) is a primitive Schur polynomial for the Abelian number field K, then s t divides the conductor of K. Moreover, s t has distinct prime factors. These results lead to a complete characterization of all primitive Schur polynomials for Abelian number fields. For arbitrary K the following elegant characterization of finite Schur polynomials of prime degree p is given: \(x^ p\) is a finite Schur polynomial for K if and only if K contains the pth cyclotomic field \(Q_ p\), and \(g…
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 3845699
zbMATH Keywords: permutation polynomials
zbMATH Keywords: characterization for Abelian number fields
zbMATH Keywords: cyclic polynomials
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Schur property
zbMATH Keywords: Schur polynomial
author: Rex W. Matthews
MaRDI profile type: Publication
cites work: Q3251484
cites work: On a conjecture of Schur
cites work: Galois Groups and Complex Multiplication
cites work: Class field theory summarized
cites work: Q4769119
cites work: Permutation polynomials over rings of algebraic integers
cites work: Q5566639
full work available at URL: 0022-314x(84)90060-x
OpenAlex ID: W1984459642
description: scientific article; zbMATH DE number 3845699