Editing Normal basis (section)

== Group representation point of view ==
A field extension {{nowrap|''K'' / ''F''}} with Galois group ''G'' can be naturally viewed as a [[Group representation|representation]] of the group ''G'' over the field ''F'' in which each automorphism is represented by itself.  Representations of ''G'' over the field ''F'' can be viewed as left modules for the [[Group ring|group algebra]] ''F''[''G'']. Every homomorphism of left ''F''[''G'']-modules <math>\phi:F[G]\rightarrow K</math> is of form <math>\phi(r) = r\beta</math> for some <math>\beta \in K</math>. Since <math>\{1\cdot \sigma| \sigma \in G\}</math> is a linear basis of ''F''[''G''] over ''F'', it follows easily that <math>\phi</math> is bijective iff <math>\beta</math> generates a normal basis of ''K'' over ''F''. The normal basis theorem therefore amounts to the statement saying  that if {{nowrap|''K'' / ''F''}} is finite Galois extension, then <math>K \cong F[G]</math> as a left <math>F[G]</math>-module. In terms of representations of ''G'' over ''F'', this means that ''K'' is isomorphic to the [[regular representation]].