On canonical roots of fractional ideals

Abstract

We give an algorithm to compute in polynomial time the roots of a fractional ideal of an order $R$. We take care not to assume $R$ is Dedekind, since the maximal order of a number field is generally inaccessible in polynomial time. Consequently, the output of such an algorithm is no longer uniquely defined. For it to be a satisfying algorithm we additionally require it be functorial, i.e., isomorphisms on the inputs should induce isomorphisms on the outputs. To adhere to these two constraints, we generalize results from Dade--Taussky--Zassenhaus, and Ge and Buchmann--Eisenbrand.