  Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$.
While much is known about the generating sets of reductions of ideals of $R$ if
$k$ is infinite, the case in which $k$ is finite is less well understood. We
investigate the existence (or lack thereof) of proper reductions of an ideal of
$R$ and the number of generators needed for a reduction in the case $k$ is a
finite field. When $R$ is one-dimensional, we give a formula for the smallest
integer $n$ for which every ideal has an $n$-generated reduction. It follows
that in a one-dimensional local Noetherian ring every ideal has a principal
reduction if and only if the number of maximal ideals in the normalization of
the reduced quotient of $R$ is at most $|k|$. In higher dimensions, we show
that for any positive integer, there exists an ideal of $R$ that does not have
an $n$-generated reduction and that if $n \geq \dim R$ this ideal can be chosen
to be $\mathfrak{m}$-primary. In the case where $R$ is a two-dimensional
regular local ring, we construct an example of an integrally closed
$\mathfrak{m}$-primary ideal that does not have a $2$-generated reduction and
thus answer in the negative a question raised by Heinzer and Shannon.
