On some subgroups of the multiplicative group of finite rings

José Felipe Voloch

doi:10.5802/jtnb.445

José Felipe Voloch ¹

¹ Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA

Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 233-239.

Abstract
Résumé

Let $S$ be a subset of $F_{q}$ , the field of $q$ elements and $h \in F_{q} [x]$ a polynomial of degree $d > 1$ with no roots in $S$ . Consider the group generated by the image of ${x - s ∣ s \in S}$ in the group of units of the ring $F_{q} [x] / (h)$ . In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.

Soit $S$ un sous-ensemble de $F_{q}$ , le corps à $q$ éléments et $h \in F_{q} [x]$ un polynôme de degré $d > 1$ sans racines dans $S$ . On considère le groupe généré par l’image de ${x - s ∣ s \in S}$ dans le groupe des unités de l’anneau $F_{q} [x] / (h)$ . Dans cet article nous présentons les bornes inférieures pour le cardinal de ce groupe. Notre motivation principale est une application au nouvel algorithme polynomial pour tester la primalité [AKS]. Ces bornes ont également des applications à la théorie des graphes et pour majorer le nombre de points rationnels sur les revètement abeliens de la droite projective sur les corps finis.

MR: 2145584 | Zbl: 1078.11069

DOI: 10.5802/jtnb.445

Author's affiliations:

José Felipe Voloch ¹

¹ Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA

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José Felipe Voloch. On some subgroups of the multiplicative group of finite rings. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 233-239. doi : 10.5802/jtnb.445. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.445/

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