On Euclidean systems of ray classes

Abstract

Lenstra introduced the notion of Euclidean ideal classes, and Treatman extended it to Euclidean systems. In this paper, we formulate Euclidean systems for ray classes, and study their basic properties. In particular, we show that every Euclidean system of ray classes generates the corre sponding ray class group. We further prove, assuming GRH, that if $K$ is a totally real Galois number field of degree $n\ge 3$ and $p$ is an odd ratio nal prime which does not split completely in $K$, then for every $N>0$, every generating set of the ray class group $Cl_K^{(p)^N}$ with modulus $(p)^N$ is a Euclidean system.