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A NEW PROOF OF THE CARLITZ–LUTZ THEOREM

  • RACHID BOUMAHDI (a1), OMAR KIHEL (a2), JESSE LARONE (a3) and MAKHLOUF YADJEL (a4)

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.

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The second and third authors were supported by NSERC.

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[1] Ayad, M., Belghaba, K. and Kihel, O., ‘On permutation binomials over finite fields’, Bull. Aust. Math. Soc. 89(1) (2014), 112124.
[2] Carlitz, L. and Lutz, J. A., ‘A characterization of permutation polynomials over a finite field’, Amer. Math. Monthly 85 (1978), 746748.
[3] Dickson, L. E., Linear Groups with an Exposition of the Galois Field Theory (Dover, New York, 1958).
[4] Lidl, R. and Mullen, G. L., ‘Does a polynomial permute the elements of the field?’, Amer. Math. Monthly 95 (1988), 243246.
[5] Lidl, R. and Niedereiter, H., Finite Fields, Encyclopedia of Mathematics and its Applications, 20 (Cambridge University Press, Cambridge, 2008).
[6] Macdonald, I. G., Symmetric Functions and Hall Polynomials (Clarendon Press, Oxford, 1998).
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A NEW PROOF OF THE CARLITZ–LUTZ THEOREM

  • RACHID BOUMAHDI (a1), OMAR KIHEL (a2), JESSE LARONE (a3) and MAKHLOUF YADJEL (a4)

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