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

We present algorithms for constructing and tabulating degree- dihedral extensions of Fq(x), where q 1 mod 2. 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- 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.

function field, Galois group, dihedral extension, construction, tabulation
Mathematical Subject Classification 2010
Primary: 11R58
Secondary: 11Y40
Published: 14 November 2013
Colin Weir
Department of Mathematics
Simon Fraser University
8888 University Drive
Burnaby, BC  V5A 1S6
Renate Scheidler
Department of Mathematics and Statistics
University of Calgary
2500 University Drive NW
Calgary, AB  T2N 1N4
Everett W. Howe
Center for Communications Research
4320 Westerra Court
San Diego, CA 92121-1969
United States