Primitive element theorem: Difference between revisions

Revision as of 21:48, 19 December 2023

Revision as of 21:49, 19 December 2023

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Let <math>E/F</math> be a ''[[field extension]]''. An element <math>\alpha\in E</math> is a ''primitive element'' for <math>E/F</math> if <math>E=F(\alpha),</math> i.e. if every element of <math>E</math> can be written as a [[rational function]] in <math>\alpha</math> with coefficients in <math>F</math>. If there exists such a primitive element, then <math>E/F</math> is referred to as a ''[[simple extension]]''.

If the field extension <math>E/F</math> has primitive element <math>\alpha</math> and is of finite [[Degree of a field extension|degree]] 4 <math>n = [E:F]</math>, then every element ''γ'' of ''E'' can be written in the form

:<math>\gamma =a_0+a_1{\alpha}+\cdots+a_{n-1}{\alpha}^{n-1}, </math>