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# An Explicit Treatment of Cubic Function Fields with Applications

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Published:2010-01-26
Printed: Aug 2010
• E. Landquist,
Department of Mathematics, Kutztown University of Pennsylvania, Kutztown, PA 19530, USA
• P. Rozenhart,
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4
• R. Scheidler,
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4
• J. Webster,
Department of Mathematics, Bates College, Lewiston, ME 04240, USA
• Q. Wu,
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4
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## Abstract

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
 Keywords: cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class number
 MSC Classifications: 14H05 - Algebraic functions; function fields [See also 11R58] 11R58 - Arithmetic theory of algebraic function fields [See also 14-XX] 14H45 - Special curves and curves of low genus 11G20 - Curves over finite and local fields [See also 14H25] 11G30 - Curves of arbitrary genus or genus \$ 11R16 - Cubic and quartic extensions 11R29 - Class numbers, class groups, discriminants

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