Normal Bases in Galois Theory
In Galois theory it is shown that for a field L, and a finite group G of automorphisms of L, the fixed field K of G has = |G|. In fact we can say more: L viewed as a K-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-isomorphism from L to K. 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 integers they contain. One can see already in the case of the Gaussian integers that such bases may not exist: a + bi and a − bi can never form a Z-module basis of Z because 1 cannot be an integer combination. The reasons are studied in depth in Galois module theory.
Read more about this topic: Regular Representation
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