Mathematical Notes

, Volume 79, Issue 5–6, pp 783–796

On matrix analogs of Fermat’s little theorem

  • A. V. Zarelua


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 conjecture integer matrix Fermat’s little theorem algebraic integers trace congruence Newton—Girard coefficient Galois extension 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. I. Arnold, “Fermat-Euler dynamic dynamical system and the statistics of the arithmetic of geometric progressions,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 37 (2003), no. 1, 1–18.MATHCrossRefGoogle Scholar
  2. 2.
    V. I. Arnold, “Topology of algebra: the combinatorics of squaring,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 37 (2003), no. 3, 20–35.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    V. I. Arnold, Euler Groups and the Arithmetic of Geometric Progressions [in Russian], MCCME, Moscow, 2003.Google Scholar
  4. 4.
    V. I. Arnold, “Topology and statistics of formulas of arithmetic,” Uspekhi Mat. Nauk [Russian Math. Surveys], 58 (2003), no. 4, 3–28.MATHMathSciNetGoogle Scholar
  5. 5.
    V. I. Arnold, “Fermat dynamics, the arithmetic of matrices, a finite circle, and a finite Lobachevski plane,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 38 (2004), no. 1, 1–15.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    V. I. Arnold, “Euler—Fermat matrix theorem,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 68 (2004), no. 6, 61–70.MATHMathSciNetGoogle Scholar
  7. 7.
    V. I. Arnold, “Geometry and dynamics of Galois fields,” Uspekhi Mat. Nauk [Russian Math. Surveys], 59 (2004), no. 6, 23–40.MATHMathSciNetGoogle Scholar
  8. 8.
    V. I. Arnold, “Number-theoretic turbulence in Fermat—Euler arithmetics and large Young diagrams geometry statistics,” J. Math. Fluid Mech., 7 (2005), S4–S50.MATHCrossRefGoogle Scholar
  9. 9.
    V. I. Arnold, “Ergodic and arithmetical properties of geometric progressions dynamics,” Moscow Math. J. (2005 (to appear)).Google Scholar
  10. 10.
    V. I. Arnold, “On the matricial version of Fermat—Euler congruences,” Japanese J. Math. (2005 (to appear)).Google Scholar
  11. 11.
    Algebraic Number Theory (J. W. S. Cassels and A. Froehlich, Eds.) Academic Press, London, 1967.MATHGoogle Scholar
  12. 12.
    Z. I. Borevich and I. R. Shafarevich, Number Theory [in Russian], Nauka, Moscow, 1985.MATHGoogle Scholar
  13. 13.
    S. Lang, Algebraic numbers, Addison-Wesley, Reading Mass., 1964.MATHGoogle Scholar
  14. 14.
    J.-P. Serre, Corps locaux, Hermann, Paris, 1962.MATHGoogle Scholar
  15. 15.
    O. Zariski and P. Samuel, Commutative Algebra. vol. 1, 2, D. Van Nostrand Co. Inc., Princeton, 1958, 1960.MATHGoogle Scholar
  16. 16.
    T. Schönemann, “Grundzüge einer allgemeinen Theorie der höhern Congruenze, deren Modul eine reelle Primzahl ist,” J. Reine Angew. Math., 31 (1846), 269–325.MATHGoogle Scholar
  17. 17.
    S. J. Smyth, “A coloring proof of a generalization of Fermat’s Little Theorem,” Amer. Math. Monthly, 93 (1986), no. 6, 469–471.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    T. Szele, “Une généralisation de la congruence de Fermat,” Matematisk tidsskrift, B (1948), 57–59.Google Scholar
  19. 19.
    L. E. Dickson, History of the Theory of Numbers. V. 1, Chelsea, New York, 1971.Google Scholar
  20. 20.
    V. V. Prasolov, Polynomials [in Russian], MCCME, Moscow, 2003.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

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

Personalised recommendations