[FOM] Need reference for results in Field Theory

Frank Waaldijk frank.waaldijk at hetnet.nl
Thu Oct 19 06:10:27 EDT 2006

A reply to Joe Shipman's questions on elementary equivalence classes of 
fields of characteristic 0 (for the questions see below):

2a. I would think so. For instance I would estimate Q to be elementarily 
equivalent to Q(X). This estimate is based on the observation that when 
playing the Fraissé-Ehrenfeucht game between Q and Q(X) (say of length n), 
one can always find for example a very large prime which satisfies the same 
`n-transcendency' as X. Therefore one has a winning strategy in this game, 
implying the two fields are elementarily equivalent. (But I did not write 
this down as a rigourous proof.)

2b.  Yes. This answer is a bit more involved. In your list you forgot the 
algebraic closures of the transcendental extensions. So let's take a look 
and let F be a countable field of characteristic 0. It is specifiable in the 
first-order language what the intersection of F is with A (field of 
algebraic numbers). Now consider subfields of the algebraic closure of A(X) 
. More specific: consider the subfield B which is obtained by repeatedly 
adding square roots of elements of A(X), and then of these elements, etc. 
until every element of B has a square root in B. One might need transfinite 
induction for this formulation, but in the end it is all a subfield of the 
countable algebraic closure of  A(X), so the result is a countable extension 
of A(X).

However, since all the extensions involved are of degree 2, the field B does 
not contain the third-root of X. Therefore it is not elementarily equivalent 
to A. But it is not elementarily equivalent to any of the fields you 
mentioned either. Because B contains A, which is first-order detectable, B 
should then be equivalent to a transcendental extension of A, say A(X_0, 
X_1, ....). Clearly such a transcendental extension does not satisfy the 
first-order sentence: every element has a square root.

Summing up: I do not think the transcendence degree is of the utmost 
importance. For instance the algebraic closure of A(X) is elementarily 
equivalent to A (classic result, the theory of an algebraic closed field of 
characteristic 0 is Aleph_1 categorical).

My guess would be that any field of characteristic 0 is elementarily 
equivalent to a subfield of the algebraic closure of A(X). This guess is 
once more based on observing the Fraissé-Ehrenfeucht game.

Hope to have contributed something...

Frank Waaldijk.

Joe Shipman wrote:

> What I am really asking is whether there is an easily describable set of 
> representatives of the elementary equivalence classes of fields (of 
> characteristic 0).
> One candidate for this is the set of all pure transcendental extensions 
> (of transcendence degree 0, 1, 2, ... , aleph_zero) of isomorphism classes 
> of subfields of the algebraic numbers.
> So a reformulated query is:
> 2a) Are any of those fields elementarily equivalent to each other?
> 2b) Is any field of characteristic 0 inequivalent to all of those fields?

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