  This paper provides an algorithmic generalization of Dickson's method of
classifying permutation polynomials (PPs) of a given degree $d$ over finite
fields. Dickson's idea is to formulate from Hermite's criterion several
polynomial equations satisfied by the coefficients of an arbitrary PP of degree
$d$. Previous classifications of PPs of degree at most $6$ were essentially
deduced from manual analysis of these polynomial equations. However, these
polynomials, needed for that purpose when $d>6$, are too complicated to solve.
Our idea is to make them more solvable by calculating some radicals of ideals
generated by them, implemented by a computer algebra system (CAS). Our
algorithms running in SageMath 8.6 on a personal computer work very fast to
determine all PPs of degree $8$ over an arbitrary finite field of odd order
$q>8$. The main result is that for an odd prime power $q>8$, a PP $f$ of degree
$8$ exists over the finite field of order $q$ if and only if $q\leqslant 31$
and $q\not\equiv 1\ (\mathrm{mod}\ 8)$, and $f$ is explicitly listed up to
linear transformations.
