A NOTE ON FERMAT’S LITTLE

THEOREM AND INTEGER MATRICES

Geoff C. Smith

This note addresses the implicit challenge in the final sentence of [1]. In that paper it is shown that if A is a square matrix with integer entries and p is a prime number then tr(A) ≅ tr(A^{p}) mod p. Other related results are also proved.

We work with polynomials in a variable X and coefficients in the integers modulo p. Observe that if f = f(X), g are any such polynomials then both (f +g)^{p} = f^{p}+g^{p} and f(X)^{p} = f(X^{p}). Now let χ = det(X·I−A) be the characteristic polynomial of A with coefficients in the integers modulo p. Let ϕ be the corresponding characteristic polynomial for A^{p} then

ϕ(X^{p}) = det(X^{p}I − A^{p}) = χ(X)^{p} = χ(X^{p}).

Thus ϕ = χ and so all elementary symmetric polynomials in the eigenvalues of A and A^{p} co-incide modulo p. In particular, the trace of A and the trace of A^{p} are the same modulo p. If A is invertible modulo p then of course tr(A^{−1}) and tr(A^{−p}) also co-incide modulo p.

References

1. Gary J. Tee, Fermat’s little theorem generalized to algebraic integers and to integer matrices., The Australian Mathematical Society Gazette 21 (5) (1994), 160-164.

University of Melbourne, Parkville, Victoria 3052 and University of Bath, Bath BA2 7AY, England.

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