Rocky Mountain Journal of Mathematics

Non-vanishing of Carlitz-Fermat quotients modulo primes

Nguyen Ngoc Dong Quan

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Abstract

In this note, we prove several non-vanishing results of Carlitz-Fermat quotients.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 1 (2016), 125-130.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1464035855

Digital Object Identifier
doi:10.1216/RMJ-2016-46-1-125

Mathematical Reviews number (MathSciNet)
MR3506081

Zentralblatt MATH identifier
1346.11062

Subjects
Primary: 11G09: Drinfelʹd modules; higher-dimensional motives, etc. [See also 14L05] 11R58: Arithmetic theory of algebraic function fields [See also 14-XX] 11T55: Arithmetic theory of polynomial rings over finite fields

Keywords
Carlitz module Carlitz-Fermat quotients Mersenne prime Wieferich prime

Citation

Quan, Nguyen Ngoc Dong. Non-vanishing of Carlitz-Fermat quotients modulo primes. Rocky Mountain J. Math. 46 (2016), no. 1, 125--130. doi:10.1216/RMJ-2016-46-1-125. https://projecteuclid.org/euclid.rmjm/1464035855


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References

  • L.E. Dickson, History of the theory of numbers, Volume I: Divisibility and primality, Chelsea Publishing Company, New York, 1966.
  • N.N. Dong Quan, Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields, J. Num. Theor. 145 (2014), 181–193.
  • D. Goss, Basic structures of function field arithmetic, Ergeb. Math. Grenzg. 35, Springer-Verlag, Berlin, 1996.
  • D.R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77–91.
  • W. Johnson, On the nonvanishing of Fermat quotients $\pmod{p}$, J. reine angew. Math. 292 (1977), 196–200.
  • V. Mauduit, Quotients de Fermat-Carlitz, C.R. Acad. Sci. Paris 321 (1995), 1139–1141.
  • ––––, Carmichael-Carlitz polynomials and Fermat-Carlitz quotients, in Finite fields and applications, Lond. Math. Soc. Lect. Note 233, Cambridge University Press, Cambridge, 1996.
  • M. Rosen, Number theory in function fields, Grad. Texts Math. 210, Springer-Verlag, New York, 2002.
  • D.S. Thakur, Iwasawa theory and cyclotomic function fields, in Arithmetic geometry Contemp. Math. 174, American Mathematical Society, Philadelphia, 1994.
  • ––––, Function field arithmetic, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
  • ––––, Fermat versus Wilson congruences, arithmetic derivatives and zeta values, preprint, 2014.