History of the Normal Basis Theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:53:38Z http://mathoverflow.net/feeds/question/35577 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35577/history-of-the-normal-basis-theorem History of the Normal Basis Theorem Anthony Pulido 2010-08-14T13:31:54Z 2010-08-16T03:16:56Z <p>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 $F.$</p> <p>Who discovered this?</p> <p>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.</p> http://mathoverflow.net/questions/35577/history-of-the-normal-basis-theorem/35578#35578 Answer by KConrad for History of the Normal Basis Theorem KConrad 2010-08-14T14:18:54Z 2010-08-15T02:51:08Z <p>The cached page </p> <p><a href="http://webcache.googleusercontent.com/search?q=cache:q5q43iNq1SQJ:siba2.unile.it/ese/issues/1/690/Notematv27n1p5.ps+normal+basis+theorem&amp;cd=3&amp;hl=en&amp;ct=clnk&amp;gl=us&amp;client=safari" rel="nofollow">http://webcache.googleusercontent.com/search?q=cache:q5q43iNq1SQJ:siba2.unile.it/ese/issues/1/690/Notematv27n1p5.ps+normal+basis+theorem&amp;cd=3&amp;hl=en&amp;ct=clnk&amp;gl=us&amp;client=safari</a></p> <p>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).</p> <p>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.</p>