On matrix analogs of Fermat’s little theorem
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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 wordsArnold conjecture integer matrix Fermat’s little theorem algebraic integers trace congruence Newton—Girard coefficient Galois extension
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- 3.V. I. Arnold, Euler Groups and the Arithmetic of Geometric Progressions [in Russian], MCCME, Moscow, 2003.Google Scholar
- 9.V. I. Arnold, “Ergodic and arithmetical properties of geometric progressions dynamics,” Moscow Math. J. (2005 (to appear)).Google Scholar
- 10.V. I. Arnold, “On the matricial version of Fermat—Euler congruences,” Japanese J. Math. (2005 (to appear)).Google Scholar
- 18.T. Szele, “Une généralisation de la congruence de Fermat,” Matematisk tidsskrift, B (1948), 57–59.Google Scholar
- 19.L. E. Dickson, History of the Theory of Numbers. V. 1, Chelsea, New York, 1971.Google Scholar