计算群代数中阶上自由模的生成元
Computing generators of free modules over orders in group algebras
摘要：
设E是数域，G是有限群.设A是群代数E[G]中的任意满秩O_E-序，X是（左）A-格.给出了X在A上不含给定秩d的一个充要条件。在E[G]的Wedderburn分解是显式可计算的并且每个分量实际上是域上的矩阵环的情况下，这导致一种算法，该算法要么给出X的A-基，要么确定不存在这样的基。设L/K是具有伽罗瓦群G的数域的有限伽罗瓦扩张，使得E是K的子域，d=[K：E].该算法可以应用于某些伽罗瓦模，自然出现在这种情况下。例如，可以取X为O_L，即L的代数整数代数整数环，A为O_L在E[G]中的相关阶A。当K=E=Q时，在某些额外的假设下，在Magma中实现了该算法对这种特殊情况的应用。
英文摘要：
Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A of O_L in E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q.
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