Mathematical Notes

, Volume 79, Issue 5, pp 783–796

On matrix analogs of Fermat’s little theorem

  • A. V. Zarelua
Article

DOI: 10.1007/s11006-006-0090-y

Cite this article as:
Zarelua, A.V. Math Notes (2006) 79: 783. doi:10.1007/s11006-006-0090-y

Abstract

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.

Key words

Arnold conjectureinteger matrixFermat’s little theoremalgebraic integerstracecongruenceNewton—Girard coefficientGalois extension

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. V. Zarelua
    • 1
  1. 1.Moscow State Technology University “Stankin”Russia