Talk:Real closed field

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A real closed field is an ordered field F in which any of the following identical conditions are true:

(1) Every non-negative element of F has a square root in F, and any polynomial of odd degree has at least one root in F.

(2) The field extension is algebraically closed.

(3) F has no proper algebraic extension to an ordered field.

If F is any ordered field, the Artin-Schreier theorem states that F has an algebraic extension, the real closure K of F, such that K is real closed and whose ordering is an extension of the ordering on F. For example, the real closure of the rational numbers are the real algebraic numbers.

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This article desperately needs to be reviewed by someone familiar with the subject. I just removed some obvious nonsense, but there are still some highly suspicious statements (e.g., the definition of superreal field does not look to me like something which could guarantee real-closedness). EJ 22:18, 6 May 2005 (UTC)

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Moreover, we do not need ultrapowers to construct Ϝ, we can do so much more constructively as the subfield of of formal power series on the Sierpinski group with a countable number of nonzero terms.

What are Sierpinski groups, and where can I find this construction? Tlepp 09:06, 8 November 2007 (UTC)

The general construction is decribed in the last but one subsection of formal power series, but I have no idea what the Sierpiński group is. -- EJ 10:32, 8 November 2007 (UTC)

I know what a power series is. The question is: how do you select a subfield of it constructively, and prove it's F. —Preceding unsigned comment added by Tlepp (talk • contribs) 11:18, 8 November 2007 (UTC)

I am not sure I follow you. I assume that "constructively" in the description of the field just means that it is explicitly defined (as opposed to the ultrapower construction, where you need to appeal to the axiom of choice to show, nonconstructively, the existence of a nonprincipal ultrafilter). Which it indeed is, provided the Sierpiński group (whatever it may be) is explicitly defined.

As for the proof (I am guessing here): you need to show that the structure is a RCF of cardinality of the continuum with the η₁ property. Now, by the comments in the formal power series article, it is automatically an ordered field, and it is real-closed if G is divisible. Obviously, it has the cardinality of continuum if G does (this is the place where the restriction to series with countable support is vital). It thus suffices to show the η₁ property, which I think should follow with a bit of effort from the η₁ property of G.

Incidentally, the conditions on G mentioned above are easily seen to be necessary, hence we can actually deduce what the Sierpiński group is: it is the divisible totally ordered abelian group of cardinality of the continuum with the η₁ property. (Such a structure is unique up to isomorphism, assuming the continuum hypothesis.) How to construct it explicitly is another matter though. -- EJ 11:11, 9 November 2007 (UTC)

Thanks for help, now it seems plausible with Sierpiński group hypothesis Tlepp 14:05, 9 November 2007 (UTC)