Arne Ledet - Some small 2-groups as Galois groups

In 1936, Witt gave a criterion for embedding a V₄-extension into a Q₈-extension in terms of an equivalence of quadratic forms. Using this result, it is possible to give similar criteria for solving other embedding problems, starting with embedding a D₄-extension into a QD₈-extension, where D₄ and QD₈ are the dihedral and quasi-dihedral groups of order 8 and 16, resp. This again permits a complete description of extensions with certain 2-groups of order 16 as Galois groups.