From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3302 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 09:45:45 -0400 (EDT) Message-ID: <200605051345.k45DjjeM026892@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241019216 8156 80.91.229.2 (29 Apr 2009 15:33:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:36 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri May 5 16:01:52 2006 -0300 X-Keywords: X-UID: 222 Original-Lines: 29 Xref: news.gmane.org gmane.science.mathematics.categories:3302 Archived-At: Mike asks: Is there a reference for the fact that a countable decidably ordered field has a constructable (and decidably ordered) real closure? The expert on all such questions is Anil Nerode .See his Effective content of field theory (Ann. Math. Logic 17 (1979), no. 3, 289--320) for a collection of all the relevant results. John writes: Briefly, while the existence of an algebraic closure of Q can be shown without choice, it uniqueness-up-to-isomorphism seems to require choice. Also, while arithmetic operations in Qbar are computable, they seem to present interesting challenges. The need for choice could hardly arise when working with a decidable countable structure such as Q. One way of naming an algebraic real number is with an ordered triple , where l and r are rationals, P a monic polynomial with rational coeficients that is irreducible over the rationals (the decidablity of which can be found in van der Waerden) such that P(l)P(r) < 0 and R(l)R(r) > 0 for R any of the non-tivial iterated derivatives of P. Another such triple names the same element iff the polynomials are the same and the intervals overlap. Effective constructions for the ordered-field operations in this context are pretty standard.