Constructing and tabulating dihedral function fields

We present algorithms for constructing and tabulating degree-` dihedral extensions of Fq(x), where q ≡ 1 mod 2`. 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-` dihedral extensions of Fq(x) up to a given discriminant bound, and we present tabulation data. We also give a formula for the number of degree-` dihedral extensions of Fq(x) with discriminant divisor of degree 2(` − 1), the minimum possible.