From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3305 Path: news.gmane.org!not-for-mail From: Peter Freyd Newsgroups: gmane.science.mathematics.categories Subject: Re: Construction of a real closure Date: Fri, 5 May 2006 16:34:50 -0400 (EDT) Message-ID: <24080.986940726$1241019218@news.gmane.org> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241019218 8165 80.91.229.2 (29 Apr 2009 15:33:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:38 +0000 (UTC) To: baez@math.ucr.edu, categories@mta.ca.barr.anil@math.cornell.edu Original-X-From: rrosebru@mta.ca Sat May 6 11:43:58 2006 -0300 X-Keywords: X-UID: 225 Original-Lines: 17 Xref: news.gmane.org gmane.science.mathematics.categories:3305 Archived-At: If memory serves me, Nerode proved years ago that the necessary and sufficient condition that all the algebraic closures of a field are effectively isomorphic is that its polynomials effectively factor into irreducibles. As for the later, the ancient theorem is that if a unique factorization domain has only finitely many units and if the factorization is effective then so it is for its polynomial ring. The condition on units pretty much forces the proof.