Ideal arithmetic and infrastructure in purely cubic function fields

Scheidler, Renate

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 609-631.

Dans cet article, nous étudions l’arithmétique des idéaux fractionnnaires dans les corps de fonctions cubiques purs, ainsi que l’infrastructure de la classe des idéaux principaux lorsque le groupe des unités du corps est de rang $1$. Nous décrivons d’abord la décomposition des polynômes irréductibles dans l’ordre maximal du corps. Nous construisons ensuite des bases d’idéaux, dites canoniques, bien adaptées pour les calcul. Nous énonçons des algorithmes permettant de multiplier les idéaux, et même de les réduire lorsque le groupe des unités est de rang $1$ et la caractéristique au moins $5$, L’article se termine avec une analyse de l’infrastructure de l’ensemble des idéaux fractionnaires réduits principaux dans le cas des corps cubiques purs de groupe des unités de rang $1$ et de caractéristique au moins $5$.

This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least five, ideal reduction. The paper concludes with an analysis of the infrastructure in the set of reduced fractional principal ideals of a purely cubic function field of unit rank one and characteristic at least five.

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