#### Vol. 1, No. 1, 2013

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Constructing and tabulating dihedral function fields

### Colin Weir, Renate Scheidler and Everett W. Howe

Vol. 1 (2013), No. 1, 557–585
##### Abstract

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 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.

##### Keywords
function field, Galois group, dihedral extension, construction, tabulation
Primary: 11R58
Secondary: 11Y40
##### Milestones
Published: 14 November 2013
##### Authors
 Colin Weir Department of Mathematics Simon Fraser University 8888 University Drive Burnaby, BC  V5A 1S6 Canada Renate Scheidler Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary, AB  T2N 1N4 Canada Everett W. Howe Center for Communications Research 4320 Westerra Court San Diego, CA 92121-1969 United States