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La théorie d’un anneau de polynomes. (French) Zbl 0563.12027

We give four possible definitions of the weak second order theory of a model and prove the interpretability of one in another. We prove the existence of \(2^{\aleph_ 0}\) countable fields having the same weak second order theory. We give sketches of proofs for the following: for any nontrivial commutative totally orderable monoid G, the weak second order theory of a field F is interpretable in the theory of the polynomial ring F[G], uniformly for any infinite field F; the same and the converse interpretabilities are true for all fields, under some more conditions on G. We end with some results about the models of the theory of a polynomial ring. More detailed proofs will appear in J. Symb. Logic.

MSC:

12L12 Model theory of fields
03B15 Higher-order logic; type theory (MSC2010)
03C60 Model-theoretic algebra
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References:

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[2] A. BAUVAL : La th\?ie du premier ordre des anneaux de polynô-\? EUR sur des corps , th\? de 3^\circ cycle, Université <e Paris VII, 1983 .
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