Comments on: Artin-Zorn theorem http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/ Dedicated to the mathematical arts. Sun, 20 Jul 2008 20:02:33 +0000 http://wordpress.org/?v=2.5.1 By: Walt http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58157 Walt Mon, 07 Jan 2008 05:51:33 +0000 http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58157 Todd, I fixed your original comment. Todd, I fixed your original comment.

]]> By: Todd Trimble http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58150 Todd Trimble Sun, 06 Jan 2008 03:12:12 +0000 http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58150 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, <a href="http://books.google.com/books?id=oyxnWF9ssI8C&pg=PA375&lpg=PA375&dq=%22artin+zorn%22&source=web&ots=xUZMl8O3yJ&sig=fmB0sadY7yFJ4dtDzsdkog_QARk" rel="nofollow">here</a>. 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.

]]> By: Todd Trimble http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58149 Todd Trimble Sun, 06 Jan 2008 03:07:24 +0000 http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58149 You can also find an online reference to Artin-Zorn as a Google Book result (Hall's Theory of Groups), with a proof, <a href="http://books.google.com/books?id=oyxnWF9ssI8C&pg=PA375&lpg=PA375&dq=%22artin+zorn%22&source=web&ots=xUZMl8O3yJ&sig=fmB0sadY7yFJ4dtDzsdkog_QARk" rel="nofollow">here</a>. 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. 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.

]]> By: Walt http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58033 Walt Sun, 23 Dec 2007 03:45:33 +0000 http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58033 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. 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.

]]> By: John Baez http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58031 John Baez Sun, 23 Dec 2007 01:04:58 +0000 http://www.arsmathematica.net/archives/2007/12/20/artin-zorn-theorem/#comment-58031 Truly this blog covers math from A to Z! Truly this blog covers math from A to Z!

]]>