Editing Normal basis (section)

== Normal basis theorem ==
Let <math>F\subset K</math> be a Galois extension with Galois group <math>G</math>. The classical '''normal basis theorem''' states that there is an element <math>\beta\in K</math> such that <math>\{g(\beta) : g\in G\}</math> forms a basis of ''K'', considered as a vector space over ''F''. That is, any element <math>\alpha \in K</math> can be written uniquely as <math display="inline">\alpha = \sum_{g\in G} a_g\, g(\beta)</math> for some elements <math>a_g\in F.</math>

A normal basis contrasts with a [[Primitive element theorem|primitive element]] basis of the form <math>\{1,\beta,\beta^2,\ldots,\beta^{n-1}\}</math>, where <math>\beta\in K</math> is an element whose minimal polynomial has degree <math>n=[K:F]</math>.