[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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