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\title
A NOTE ON FERMAT'S LITTLE THEOREM AND INTEGER MATRICES
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\author
Geoff C. Smith
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\address
University of Melbourne,
Parkville,
Victoria 3052
and University of Bath, Bath BA2 7AY, England.
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This note addresses the implicit challenge in the final sentence
of \cite{1}. In that paper
it is shown that if $A$ is a square matrix with integer entries
and $p$ is a prime number then $tr(A) \cong tr(A^p) \hbox{ mod } p.$
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 $\chi = \hbox{det}(X\cdot I - A)$ be the characteristic
polynomial of $A$ with coefficients in the integers modulo $p.$
Let $\phi$ be the corresponding characteristic polynomial for $A^p$ then
$$\phi(X^p) = \hbox{det}(X^p I - A^p) = \chi(X)^p = \chi(X^p).$$
Thus $\phi = \chi$ 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 $p.$


\Refs

\ref\key 1
\by Gary J. Tee
\paper Fermat's little theorem generalized to algebraic integers and to
integer matrices.
\jour The Australian Mathematical Society Gazette
\vol 21 (5)
\yr 1994
\pages 160-164
\endref

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