# 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

The cached page

gives some information: Eisenstein conjectured it in 1850 for extensions of finite fields and Hensel gave a proof for finite fields in 1888. Dedekind used such bases in number fields in his work on the discriminant in 1880, but he had no general proof. (See the quote by Dedekind on the bottom of page 51 of Curtis's "Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer".) In 1932 Noether gave a proof for some infinite fields while Deuring gave a uniform proof for all fields (also in 1932).

In Narkiewicz's "Elementary and Analytic Theory of Algebraic Numbers" (3rd ed.) he writes on the bottom of p. 186 that the normal basis theorem is due to Noether, but as usual the history is slightly more complicated.

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My, my, it was right there on the first page! And thank you also for the other information. I looked at Narkiewicz's book on Google books just now. This helps a great deal. Many thanks again. – Anthony Pulido Aug 14 '10 at 14:53
Anthony: it was right there on the first page for what search terms? (Of course now if you search for "normal basis theorem history" the first google hit is this MO page, but I don't think that's what you meant.) – KConrad Aug 15 '10 at 2:49
Oh, I certainly don't mean to disparage your search abilities. I was merely expressing my embarrassment. If you use the search terms "normal basis theorem," that paper is one of the first results. Originally, the first sentence in my comment read "...and I missed it completely!" but I thought I'd minimize the embarrassment to myself. I did find that paper early on, and actually, I printed it out and it has been sitting on my bookshelf for some time. I really should have looked at it more carefully, because the answer was on the very first page. Again, many, many thanks for your trouble. – Anthony Pulido Aug 15 '10 at 4:57
Yes, thank you for pointing out Curtis's book, a wonderful source, which originally lead me to the Collected Works. I looked at both of Hawkins's papers, but I didn't find much there, although the sections on the Dedekind-Frobenius correspondence was fascinating. Have you looked at Roggenkamp? It's possible I've already done so, but I'll look at it again when I'm in the library. – Anthony Pulido Aug 16 '10 at 3:35
You won't find anything new about the normal basis theorem in the article by Roggenkamp (On Dedekind's group determinant and Frobenius' higher characters). He writes that Dedekind was looking at the discriminant of a basis of a Galois extension of number fields (or just of Q?) and considered the case of a normal basis, although the existence of such a basis in general was only proved later by others. This is already in the other references that have been mentioned above. – KConrad Aug 16 '10 at 3:52