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Constructing and tabulating dihedral function fields. (English) Zbl 1345.11079
Howe, Everett W. (ed.) et al., ANTS X. Proceedings of the tenth algorithmic number theory symposium, San Diego, CA, USA, July 9–13, 2012. Berkeley, CA: Mathematical Sciences Publishers (MSP) (ISBN 978-1-935107-00-2/hbk; 978-1-935107-01-9/ebook). The Open Book Series 1, 557-585 (2013).
Summary: We present algorithms for constructing and tabulating degree-$$\ell$$ dihedral extensions of $$\mathbb F_q(x)$$, where $$q\equiv 1\mod 2\ell$$. 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 $$\mathbb F_q(x)$$ 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 $$\mathbb F_q(x)$$ with discriminant divisor of degree $$2(\ell -1)$$, the minimum possible.
For the entire collection see [Zbl 1295.11003].

##### MSC:
 11R58 Arithmetic theory of algebraic function fields 11Y40 Algebraic number theory computations
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