Geodesic
On matrix analogs of Fermat's little theorem
Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 838-853

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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. 
@article{MZM_2006_79_6_a2,
     author = {A. V. Zarelua},
     title = {On matrix analogs of {Fermat's} little theorem},
     journal = {Matemati\v{c}eskie zametki},
     pages = {838--853},
     publisher = {mathdoc},
     volume = {79},
     number = {6},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_6_a2/}
}
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A. V. Zarelua. On matrix analogs of Fermat's little theorem. Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 838-853. http://geodesic.mathdoc.fr/item/MZM_2006_79_6_a2/