Computing the multiplicative group of residue class rings

Abstract:

Let $\mathbf {k}$ be a global field with maximal order $\mathfrak o_{\mathbf k}$ and let ${\mathfrak {m}}_{0}$ be an ideal of $\mathfrak o_{\mathbf k}$. We present algorithms for the computation of the multiplicative group $(\mathfrak o_{\mathbf k}/{\mathfrak {m}}_{0})^*$ of the residue class ring $\mathfrak o_{\mathbf k}/{\mathfrak {m}}_{0}$ and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group $\mathbf {Cl}_{\mathbf {k}}^{\mathfrak {m}}$ modulo $\mathfrak m={\mathfrak {m}}_{0}{\mathfrak {m}}_{\infty }$, where ${\mathfrak {m}}_{\infty }$ denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.