Generic Galois extensions for families of finite groups

We study the existence of generic Galois extensions for two families of finite groups. We first look at the case of central extensions of symmetric groups and we show that such groups have generic Galois extensions over any field of characteristic zero. One of the crucial tools required in this case is from the theory of quadratic forms, which enables us to solve embedding problems. Rather than solving embedding problems for one field extension at a time, we do this for an entire family of extensions. This is done by using a one-one correspondence between monic polynomials (whose splitting fields are under consideration) and points in affine space. One of the consequences of this result is an if-and-only-if criterion as to when certain families of nonabelian groups have generic Galois extensions. The other family of finite groups in our consideration are dihedral groups. By showing that the problem can be understood by looking at the case of dihedral 2-groups and dihedral groups of order 2n for n odd, we reduce the problem to easier problems. We then discuss a proof for the case of 2 n with n odd. We also give an example for the case of the dihedral group of order 8.