Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Constructing and tabulating dihedral function fields

Renate Scheidler and Colin Weir

Abstract: We present algorithms for constructing and tabulating degree l dihedral function fields over a finite field F_q of odd characteristic with q congruent to 1 modulo l. 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 the main theorem, which gives an exact count for such fields. We then use this construction method in a tabulation algorithm to construct all cubic function fields over F_q up to a given discriminant bound, and provide tabulation data.

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