Primitive element theorem: Difference between revisions

Revision as of 22:02, 1 May 2021

Revision as of 22:03, 1 May 2021

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:Every [[Separable extension|separable]] field extension of finite degree is simple.

This theorem, by [[Ernst Steinitz]] (1910), applies to [[algebraic number field]]s, i.e. finite extensions of the rational numbers '''Q''', since '''Q''' has [[characteristic (algebra)|characteristic]] 0 and therefore every finite extension over '''Q''' is separable.

The following primitive element theorem is more general ([[Ernst Steinitz]], 1910):

:The finite field extension <math>E/F</math> is simple if and only if there exist only finitely many intermediate fields ''K'' with <math>E\supseteq K\supseteq F</math>.