# History of the Normal Basis Theorem

The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space over $F.$

Who discovered this?

I've looked through the collected works of Frobenius and Dedekind, which are the earliest works I've seen referring to it, but it looks like the theorem led Dedekind to what is called the group determinant, and he doesn't give a reference. (p. 433 of Dedekind's Gesammelte Werke, v. 2, via Curtis's Pioneers of Representation Theory. See KConrad's answer below.) Among others, I've also looked at some of the correspondence of Hasse and Noether. The works are in German, which is second language to me, so it's possible I missed something. Needless to say, I've searched using Google to no avail. If anyone knows something, I'd be very grateful.

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You do not need the normal basis theorem to construct the group determinant, since that polynomial det(x_{gh^{-1}}) in variables indexed by the group does not mention Galois extensions of fields. Rather, the conclusion of the normal basis theorem provided Dedekind with the motivation leading to the group determinant. What motivates a concept should be distinguished from what is logically needed to define it. –  KConrad Aug 14 '10 at 18:25
Noted, thank you very much. –  Anthony Pulido Aug 14 '10 at 20:08
I just realized I should correct the above according to your comments. Done. Many thanks again. –  Anthony Pulido Aug 16 '10 at 3:38

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