On canonical roots of fractional ideals

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.