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.
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References
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.
V. I. Arnold, “Topology of algebra: the combinatorics of squaring,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 37 (2003), no. 3, 20–35.
V. I. Arnold, Euler Groups and the Arithmetic of Geometric Progressions [in Russian], MCCME, Moscow, 2003.
V. I. Arnold, “Topology and statistics of formulas of arithmetic,” Uspekhi Mat. Nauk [Russian Math. Surveys], 58 (2003), no. 4, 3–28.
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.
V. I. Arnold, “Euler—Fermat matrix theorem,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 68 (2004), no. 6, 61–70.
V. I. Arnold, “Geometry and dynamics of Galois fields,” Uspekhi Mat. Nauk [Russian Math. Surveys], 59 (2004), no. 6, 23–40.
V. I. Arnold, “Number-theoretic turbulence in Fermat—Euler arithmetics and large Young diagrams geometry statistics,” J. Math. Fluid Mech., 7 (2005), S4–S50.
V. I. Arnold, “Ergodic and arithmetical properties of geometric progressions dynamics,” Moscow Math. J. (2005 (to appear)).
V. I. Arnold, “On the matricial version of Fermat—Euler congruences,” Japanese J. Math. (2005 (to appear)).
Algebraic Number Theory (J. W. S. Cassels and A. Froehlich, Eds.) Academic Press, London, 1967.
Z. I. Borevich and I. R. Shafarevich, Number Theory [in Russian], Nauka, Moscow, 1985.
S. Lang, Algebraic numbers, Addison-Wesley, Reading Mass., 1964.
J.-P. Serre, Corps locaux, Hermann, Paris, 1962.
O. Zariski and P. Samuel, Commutative Algebra. vol. 1, 2, D. Van Nostrand Co. Inc., Princeton, 1958, 1960.
T. Schönemann, “Grundzüge einer allgemeinen Theorie der höhern Congruenze, deren Modul eine reelle Primzahl ist,” J. Reine Angew. Math., 31 (1846), 269–325.
S. J. Smyth, “A coloring proof of a generalization of Fermat’s Little Theorem,” Amer. Math. Monthly, 93 (1986), no. 6, 469–471.
T. Szele, “Une généralisation de la congruence de Fermat,” Matematisk tidsskrift, B (1948), 57–59.
L. E. Dickson, History of the Theory of Numbers. V. 1, Chelsea, New York, 1971.
V. V. Prasolov, Polynomials [in Russian], MCCME, Moscow, 2003.
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Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840–855.
Original Russian Text Copyright © 2006 by A. V. Zarelua.
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Zarelua, A.V. On matrix analogs of Fermat’s little theorem. Math Notes 79, 783–796 (2006). https://doi.org/10.1007/s11006-006-0090-y
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DOI: https://doi.org/10.1007/s11006-006-0090-y