We present algorithms for constructing and tabulating
degree$\ell $ dihedral
extensions of
${\mathbb{F}}_{q}\left(x\right)$,
where
$q\equiv \mathfrak{1}\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}\mathfrak{2}\ell $.
We begin with a Kummertheoretic algorithm for constructing these function fields
with prescribed ramification and fixed quadratic resolvent field. This algorithm is
based on the proof of our main theorem, which gives an exact count for such fields.
We then use this construction method in a tabulation algorithm to construct all
degree$\ell $ dihedral
extensions of
${\mathbb{F}}_{q}\left(x\right)$ up to a
given discriminant bound, and we present tabulation data. We also give a formula for the number of
degree$\ell $ dihedral extensions
of
${\mathbb{F}}_{q}\left(x\right)$ with discriminant
divisor of degree
$\mathfrak{2}\left(\ell \mathfrak{1}\right)$,
the minimum possible.
