Artin-Zorn theorem
December 20th, 2007 by WaltWedderburn’s theorem (proof here) states that any finite division ring is a field. Interestingly, apparently this generalizes to nonassociative division rings that are alternative. This is known as the Artin-Zorn theorem. The best online reference I could find was here.
December 22nd, 2007 at 6:04 pm
Truly this blog covers math from A to Z!
December 22nd, 2007 at 8:45 pm
In one post, even. This is probably the highlight of the blog, and I should just close it down now and leave on a high note.
January 5th, 2008 at 8:07 pm
You can also find an online reference to Artin-Zorn as a Google Book result (Hall’s Theory of Groups), with a proof, here.
The basic idea is this: a finite alternative division ring D generated by two elements is associative, whence a field by Wedderburn’s theorem, but a finite field is generated by just one element. This shows that if D is generated by x1, x2, …, xn, then the first two generators can be replaced by one. Keep iterating this observation until you’ve chiseled your way down to a single generator.
January 5th, 2008 at 8:12 pm
Sorry, Walt, I screwed up the tag. I meant to have
You can also find an online reference to Artin-Zorn as a Google Book result (Hall’s Theory of Groups), with a proof, here.
January 6th, 2008 at 10:51 pm
Todd, I fixed your original comment.