Abstract

We present algorithms for constructing and tabulating degree-$\ell$ dihedral extensions of ${\mathbb{F}}_q\left(x\right)$, where $q\equiv \mathfrak{1}mod\mathfrak{2}\ell$. We begin with a Kummer-theoretic 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.

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Mathematical Subject Classification 2010

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