On the construction of primitive elements in field extensions

The subject of this article are a priori constructions of primitive elements in field extensions. Consider the totality of all separable polynomialsf of degreen over a fieldK with rootsx₁,...,x_n and prescribed Galois groupG. A vector (b₁,...,b_n)Kⁿ is called stably primitive (forG), if, for each suchf,b₁x₁+...+b_nx_n generates the splitting field off. We develop representation theoretical devices to investigate the set of stably primitive vectors geometrically. A fundamental observation is that is either very large or very small (or even empty). These two cases are illustrated by various examples. Moreover, criteria are given to decide which case holds. For a number of groups where is recognized to be small we show.