Computation of lattice isomorphisms and the integral matrix similarity problem

Abstract Let K be a number field, let A finite-dimensional -algebra, $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of and $\Lambda $ an $\mathcal {O}_{K}$ -order in . Suppose that each simple component semisimple -algebra $A/{\operatorname {J}}(A)}$ is isomorphic to matrix ring over field. Under this hypothesis on , we give algorithm that, given two -lattices X Y determines whether are and, if so, computes explicit isomorphism $X \rightarrow Y$ This reduces problem standard problems computational algebra algorithmic algebraic theory polynomial time. As application, for following long-standing problem: Given field positive integer n matrices $A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$ determine B similar return $C {GL}_{n}(\mathcal such $B= CAC^{-1}$ We examples show implementation latter {O}_{K}=\mathbb {Z}$ vastly outperforms implementations all previous algorithms, as predicted by our complexity analysis.