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==
If ''F'' is finite, then any finite extension ''E'' of ''F'' is simple, and there are also only finitely many intermediate fields. Otherwise, startingStarting with a simple 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'' generated over ''F'' by the coefficients of ''g''. Then 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.
