# [FOM] Need reference for results in Field Theory

zahidi@logique.jussieu.fr zahidi at logique.jussieu.fr
Fri Oct 20 04:37:35 EDT 2006

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> 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.)

I don't think this is true actually. Q (rational numbers) and Q(x) are not
elementary equivalent. An easy way to see this is as follows:
the Pythagors number p(K) of a field K is defined as the smallest number
of squares needed to represent any number of squares.
It follows from Langrange's theorem that the Pythagors number of Q is 4
(any rational number that is a sum of squares is positive; by Lagrange's
theorem
every positive rational number is the sum of four squares).
The Pythagoras number of Q(x) is 5.
It is easy to see that for any natural number n, the statement p=n is an
elementary statement. Hence the statement p=4 is true in Q but not in
Q(X).

> 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).

Results by Florian Pop show that for large classes of fields transcendence
degree is first-order definable and hence important for elementary
equivalence. E.g. Q(x) is never elementary equivalent to Q(x_1,...,x_n)
(field of rational functions in n variables), for any n>1.

Also, an argument using the Pythagors number can be used to show that R(x)
(where R denotes the reals) is never elementary equivalent to
R(x_1,...,x_n) (field of rational functions in n variables), for any n>1.

Karim Zahidi

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