Quadratic corestriction, C-2-embedding problems, and explicit construction

摘要

Let KIF be a quadratic extension of fields of characteristic not 2. We give a natural correspondence between cup-products A of square classes of K* and certain Galois embedding problems H --> G over F. Under this correspondence, the obstruction to the embedding problem associated to A is the corestriction from K to F of the quaternion algebra over K given by A. We reduce the explicit construction problem for such embedding problems to the construction of an F-algebra isomorphism from an algebra expressed as a tensor product of quaternion algebras to a matrix ring. We apply these results to determine the obstructions to, and a method for explicit construction for, all C-2-embedding problems extending D-4 boolean AND C-4 or D-4 boolean AND D-4, where the subdirect products are taken over a common factor group C-2. References: 24