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Uniform first-order definitions in finitely generated fields. (English) Zbl 1197.12005

The general question underlying the paper is the following: can a given property be expressed by the truth of a first-order sentence? The author focusses on finitely generated fields (i.e. fields that are finitely generated over the prime subfield) considered as first-order structures in the usual language of rings. The two main results can be stated as follow. First, there is a first-order sentence that is true for all finitely generated fields of characteristic \(0\) but false for all fields of positive characteristic. Second, given any positive integer \(n\), there is a formula with \(n\) free variables that, when interpreted in a finitely generated field \(K\), is true for elements \(x_1, \ldots , x_n \in K\) if and only if these elements are algebraically independent over the prime subfield of \(K\). The author also shows that the formulas expected in the two results above cannot be purely existential.

MSC:

12L12 Model theory of fields
11U09 Model theory (number-theoretic aspects)
14H52 Elliptic curves
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