Primitive element theorem: Difference between revisions

Revision as of 19:25, 19 July 2023

Revision as of 19:27, 19 July 2023

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This is almost immediate as a way of showing how Steinitz' result implies the classical result, and a bound for the number of exceptional ''c'' in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and ''a priori''). Therefore, in this case trial-and-error is a possible practical method to find primitive elements.

==Proof==

Starting with a simple finite extension ''E''=''F''(α), let ''f'' be the [[minimal polynomial (field theory)|minimal polynomial]] of α over ''F''. If ''K'' is an intermediate subfield, then let ''g'' be the minimal polynomial of α over ''K'', and let ''L'' be the subfield of ''K''field generated over ''F'' by the coefficients of ''g''. Then since ''L''⊆''K'', the minimal polynomial of α over ''L'' must be a multiple of ''g'', so it is ''g''; this implies that the degree of ''E'' over ''L'' is the same as that over ''K'', but since ''L''⊆''K'', this means that ''L''=''K''. Since ''g'' is a factor of ''f'', this means that there can be no more intermediate fields than factors of ''f'', so there are only finitely many.

Going in the other direction, if ''F'' is finite, then there are automatically only finitely many intermediate fields in any finite extension, so assume that ''F'' is infinite. Then, ''E'' is generated over ''F'' by a finite number of elements, so it's enough to prove that ''F''(α, β) is simple for any two elements α and β in ''E''.