  In this note, the first-order Dickson polynomials are introduced through a
particular case of the expression of the trace of the $n^{th}$ power of a
matrix in terms of powers of the trace and determinant of the matrix itself.
The technique relies on the Cayley-Hamilton theorem and its application to the
derivation of formulas due to Carlitz and to second-order Dickson polynomials
is straightforward. Finally, generalization of Dickson polynomials over finite
fields and multivariate Dickson polynomials are evoked as potential avenues of
investigation in the same framework.
