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A NEW PROOF OF THE CARLITZ–LUTZ THEOREM

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

Published online by Cambridge University Press:  10 July 2019

RACHID BOUMAHDI Open the ORCID record for RACHID BOUMAHDI [Opens in a new window] ,
OMAR KIHEL Open the ORCID record for OMAR KIHEL [Opens in a new window] ,
JESSE LARONE Open the ORCID record for JESSE LARONE [Opens in a new window]  and
MAKHLOUF YADJEL Open the ORCID record for MAKHLOUF YADJEL [Opens in a new window]
RACHID BOUMAHDI
Affiliation:
La3c Laboratory, Faculty of Mathematics, USTHB University, Algiers, Algeria email [email protected]
OMAR KIHEL
Affiliation:
Department of Mathematics, Brock University, Ontario, CanadaL2S 3A1 email [email protected]
JESSE LARONE*
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, CanadaG1V 0A6 email [email protected]
MAKHLOUF YADJEL
Affiliation:
La3c Laboratory, Faculty of Mathematics, USTHB University, Algiers, Algeria email [email protected]
*
✉[email protected]
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Abstract

A polynomial $f$ over a finite field $\mathbb{F}_{q}$ can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers $f^{e}$ for each $e$ from $1$ to $q-2$, as well as the existence of a unique solution to $f(x)=0$ in $\mathbb{F}_{q}$. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.

Keywords

permutation polynomialHermite–Dickson theorem

MSC classification

Primary: 11T06: Polynomials
Secondary: 12E20: Finite fields (field-theoretic aspects)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 1 , February 2020 , pp. 56 - 60
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

The second and third authors were supported by NSERC.

References

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