Editing Regular representation (section)

==Normal bases in Galois theory==

In [[Galois theory]] it is shown that for a field ''L'', and a finite group ''G'' of [[automorphism]]s of ''L'', the fixed field ''K'' of ''G'' has [''L'':''K''] = |''G''|. In fact we can say more: ''L'' viewed as a ''K''[''G'']-module is the regular representation. This is the content of the [[normal basis theorem]], a '''normal basis''' being an element ''x'' of ''L'' such that the ''g''(''x'') for ''g'' in ''G'' are a [[vector space]] basis for ''L'' over ''K''. Such ''x'' exist, and each one gives a ''K''[''G'']-isomorphism from ''L'' to ''K''[''G'']. From the point of view of [[algebraic number theory]] it is of interest to study ''normal integral bases'', where we try to replace ''L'' and ''K'' by the rings of [[algebraic integer]]s they contain. One can see already in the case of the [[Gaussian integer]]s that such bases may not exist: ''a'' + ''bi'' and ''a'' &minus; ''bi'' can never form a '''Z'''-module basis of '''Z'''[''i''] because 1 cannot be an integer combination. The reasons are studied in depth in [[Galois module]] theory.