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Keywords:
minimal generating set of splitting field of polynomial; cluster tower; multiple transitivity of Galois group
minimal generating set of splitting field of polynomial; cluster tower; multiple transitivity of Galois group
Summary:
A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous concept in the context of vector spaces: they can be of different cardinalities. In fact, we establish that for a certain family of polynomials over the rationals, we have minimal generating sets of all cardinalities in a certain range and that these are the only possible cardinalities for minimal generating set for such a polynomial. We also study how minimal generating sets behave under multiple transitivity of the Galois group and consequently prove the existence of polynomials with all minimal generating sets of uniformly same cardinality. We also connect minimal generating sets with the concept of root cluster tower of an irreducible polynomial introduced in M. Krithika, P. Vanchinathan (2024).
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