Primitive element theorem: Difference between revisions

Revision as of 19:33, 19 July 2023

Revision as of 19:33, 19 July 2023

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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 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 any finite extension of ''F'' is automatically simple, 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''. But, considering all fields ''F''(α + ''x'' β), where ''x'' is an element of ''F'', there are only finitely many, so there must be distinct ''x₀'' and ''x₁'' in ''F'' for which ''F''(α + ''x₀'' β) = ''F''(α + ''x₁'' β). Then simple algebra shows that ''F''(α + ''x₀'' β)=''F''(α, β).

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