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Title: Constructing and Tabulating Dihedral Function Fields
Author: Weir, Colin
Advisor: Scheidler, Renate
Keywords: Mathematics;Mathematics
Issue Date: 9-Sep-2013
Abstract: We present and implement algorithms for constructing and tabulating odd prime degree l dihedral extensions of a rational function field k_0(x), where k_0 is a perfect field with characteristic not dividing 2l. We begin with a class field theoretic construction algorithm when k0 is a finite field. We also describe modifications to this algorithm to improve the run time of its implementation. Subsequently, we introduce a Kummer theoretic algorithm for constructing dihedral function fields with prescribed ramification and fixed quadratic resolvent field, when k0 is a perfect field and not necessarily finite. We give a detailed description of this Kummer theoretic approach when k0, or a quadratic extension thereof, contains the l-th roots of unity. Avoiding the case when k0 is infinite and the quadratic resolvent field is a constant field extension of k0(x), we show that our Kummer theoretic algorithm constructs all degree l dihedral extensions of k0(x) with prescribed ramification. This algorithm is based on the proof of our main theorem, which gives an exact count of such fields. When k0 is a finite field, we implement and experimentally compare the run times of the class field theoretic and Kummer theoretic construction algorithms. We find that the Kummer theoretic approach performs better in all cases, and hence we use it later in our tabulation method. Lastly, utilizing the automorphism group of Fq(x), we present a tabulation algorithm to construct all degree l dihedral extensions of Fq(x) up to a given discriminant bound, and we present tabulation data. This data is then compared to known asymptotic predictions, and we find that these estimates over-count the number of such fields. We also give a formula for the number of degree l dihedral extensions of Fq(x) when q is plus or minus 1 mod 2l, with discriminant divisors of minimum possible degree.
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