Abstract
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 rootsx 1,...,x n and prescribed Galois groupG. A vector (b 1,...,b n )∈K n is called stably primitive (forG), if, for each suchf,b 1 x 1+...+b n x 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
.
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Girstmair, K. On the construction of primitive elements in field extensions. Monatshefte für Mathematik 98, 193–209 (1984). https://doi.org/10.1007/BF01507748
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DOI: https://doi.org/10.1007/BF01507748