  We study properties of the Golomb topology on polynomial rings over fields,
in particular trying to determine conditions under which two such spaces are
not homeomorphic. We show that if $K$ is an algebraic extension of a finite
field and $K'$ is a field of the same characteristic, then the Golomb spaces of
$K[X]$ and $K'[X]$ are homeomorphic if and only if $K$ and $K'$ are isomorphic.
