Permutation polynomials over algebraic number fields
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 \(g_ p(x)\) is a finite Schur polynomial for K if and only if K contains the maximal real subfield of \(Q_ p\). In the last section the results of the reviewer and S. K. Lo [loc. cit.] are derived from the author's general results. Further misprints: in the proof of Lemma 2.13 replace the index t/k of g by t; in line 3 of the proof of Theorem 4.1 and line 5 of the proof of Theorem 4.2 replace the index L/\({\mathbb{Q}}\) of N by K/\({\mathbb{Q}}\); in the proof of Lemma 5.1 replace S by s.
characterization for Abelian number fields
scientific article; zbMATH DE number 3845699