Abstract:
We are interested in the following question: given the separable extension
of degree 5 (with some restrictions on the charasteristic of the ground
field), how do we find a generating element whose equation is as simple as
possible? We will consider two approaches: the first one, which is close to
original argument of Hermite, is to construct suitable homogenous
$S_5$-covariant, while the second one is based on rationality questions for
some cubic surface associated with the field. The latter method includes
study of the lines on the surface and Galois group action on them.
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