On some subgroups of the multiplicative group of finite rings
Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 233-239.

Soit S un sous-ensemble de F q , le corps à q éléments et hF q [x] un polynôme de degré d>1 sans racines dans S. On considère le groupe généré par l’image de {x-ssS} 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.

Let S be a subset of F q , the field of q elements and hF q [x] a polynomial of degree d>1 with no roots in S. Consider the group generated by the image of {x-ssS} 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.

DOI : 10.5802/jtnb.445
Voloch, José Felipe 1

1 Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA
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Voloch, José Felipe. On some subgroups of the multiplicative group of finite rings. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 233-239. doi : 10.5802/jtnb.445. http://www.numdam.org/articles/10.5802/jtnb.445/

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