Fermat quotients over function fields
type: Item
instance of: scholarly article
title: Fermat quotients over function fields
zbMATH Open document ID: 0913.11052
author: Jim Sauerberg
author: Linghsueh Shu
published in: Finite Fields and their Applications
publication date: 1999-05-20T00:00:00Z
review text: The authors study the generalization of the classical Fermat quotients to rational function fields over finite fields: if \(P\) is an irreducible polynomial and \(A\) is any polynomial over \({\mathbb F}_{q}\), the Fermat quotient of \(A\) with respect to \(P\) is the polynomial \(Q_{P}(A) = {A^{\deg(P)} - A \over P}\). Several complete results about such quotients are obtained, while little is known for traditional Fermat quotients over the integers. For instance, they give the exact power of \(P\) dividing \(Q_{P}(A)\), and the exact size of the set \(\{ Q_{P}(A) : \deg A < \deg P \}\) considered modulo \(P\). The study of rational field Fermat quotients is reduced to that of quotients \({A^{\deg(P)} - A \over x^{q^{n}}-x}\), where \(n\) is a positive integer.
reviewed by: Leonardo Cangelmi
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1199962
zbMATH Keywords: Fermat quotients
zbMATH Keywords: rational function fields
zbMATH Keywords: polynomials over finite fields
MaRDI profile type: Publication
OpenAlex ID: W2026812209
cites work: Q5620640
cites work: Le Theoreme de Fermat-Goss
cites work: On Fermats Equation in C(t)
cites work: Q3134953
cites work: On the p-Divisibility of the Fermat Quotients
cites work: On the nonvanishing of Fermat quotients (mod p).
cites work: Q3315578
cites work: Q4718170
DOI: 10.1006/FFTA.1996.0165
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description: scientific article; zbMATH DE number 1199962