Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an O_K-order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left) Lambda-lattice X either computes a free basis of X over Lambda or shows that X is not free over Lambda. In the present article, we generalise this by showing that, under weaker hypotheses on A, there exists an algorithm that for two given Lambda-lattices X and Y either computes an isomorphism X -> Y or determines that X and Y are not isomorphic. The algorithm is implemented in Magma for A=Q[G], Lambda=Z[G] and Lambda-lattices X and Y contained in Q[G], where G is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of Q with Galois group isomorphic to Q_8 x C_2, the direct product of the quaternion group of order 8 and the cyclic group of order 2.

中文翻译：

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计算格之间的同构

令 K 为数域，令 A 为有限维半单 K 代数，令 Lambda 为 A 中的 O_K 阶。 之前的工作表明，在 A 的某些假设下，存在一种算法，对于给定的（左）Lambda-lattice X 要么在 Lambda 上计算 X 的自由基，要么表明 X 在 Lambda 上不是自由的。在本文中，我们通过证明在 A 的较弱假设下，存在一种算法来概括这一点，该算法对于两个给定的 Lambda 格 X 和 Y 要么计算同构 X -> Y，要么确定 X 和 Y 不是同构的。对于 A=Q[G]、L​​ambda=Z[G] 和包含在 Q[G] 中的 Lambda 格 X 和 Y，该算法在 Magma 中实现，其中 G 是满足某些假设的有限群。