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ON PERMUTATION BINOMIALS OVER FINITE FIELDS

Part of: General field theory Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  28 March 2013

MOHAMED AYAD ,
KACEM BELGHABA  and
OMAR KIHEL
MOHAMED AYAD
Affiliation:
Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, F-62228 Calais, France email ayad@lmpa.univ-littoral.fr
KACEM BELGHABA
Affiliation:
Laboratoire de Mathématiques et ses Applications, Université d’Oran, BP 15 24, Algeria email belghaba.kacem@univ-oran.dz
OMAR KIHEL*
Affiliation:
Department of Mathematics, Brock University, Ontario, Canada L2S 3A1 email okihel@brocku.ca
*
✉okihel@brocku.ca
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Abstract

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Let ${ \mathbb{F} }_{q} $ be the finite field of characteristic $p$ containing $q= {p}^{r} $ elements and $f(x)= a{x}^{n} + {x}^{m} $, a binomial with coefficients in this field. If some conditions on the greatest common divisor of $n- m$ and $q- 1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x)= a{x}^{n} + {x}^{m} $ permutes ${ \mathbb{F} }_{p} $, where $n\gt m\gt 0$ and $a\in { \mathbb{F} }_{p}^{\ast } $, then $p- 1\leq (d- 1)d$, where $d= \gcd (n- m, p- 1)$, and that this bound of $p$, in terms of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of ${ \mathbb{F} }_{q} $ from a permutation binomial over ${ \mathbb{F} }_{q} $.

Keywords

finite fieldspermutation polynomialsHermite–Dickson theorem

MSC classification

Secondary: 11T06: Polynomials 12E20: Finite fields (field-theoretic aspects)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 112 - 124
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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