×

zbMATH — the first resource for mathematics

Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials. (English) Zbl 0722.11060
Let \(f(x)\) be a polynomial with integral coefficients and degree \(n\geq 2\). Then it is shown that for any prime \(p>(n^2-3n+4)^2\) for which \(f\), considered modulo \(p\), is a permutation polynomial of degree \(n\) of the finite field \(\mathbb F_p\), there is no integer \(c\) with \(1\leq c<p\) for which \(f(x)+cx\) is also a permutation polynomial of \(\mathbb F_p\). This settles a conjecture of S. Chowla and H. Zassenhaus [Norske Vid. Selsk. Forhdl. 41, 34–35 (1968; Zbl 0186.09203)]. The proof makes clever use of the Lang-Weil theorem and of a result from the work of M. Fried [Mich. Math. J. 17, 41–55 (1970; Zbl 0169.37702)] on the resolution of Schur’s conjecture.

MSC:
11T06 Polynomials over finite fields
PDF BibTeX XML Cite
Full Text: DOI