## Duke Mathematical Journal

### Uniform first-order definitions in finitely generated fields

Bjorn Poonen

#### Abstract

We prove that there is a first-order sentence in the language of rings that is true for all finitely generated fields of characteristic $0$ and false for all fields of characteristic greater than $0$. We also prove that for each $n \in {\mathbb N}$, there is a first-order formula $\psi_n(x_1,\ldots,x_n)$ that when interpreted in a finitely generated field $K$ is true for elements $x_1,\ldots,x_n \in K$ if and only if the elements are algebraically dependent over the prime field in $K$

#### Article information

Source
Duke Math. J. Volume 138, Number 1 (2007), 1-21.

Dates
First available: 9 May 2007

http://projecteuclid.org/euclid.dmj/1178738559

Digital Object Identifier
doi:10.1215/S0012-7094-07-13811-0

Mathematical Reviews number (MathSciNet)
MR2309154

Zentralblatt MATH identifier
05166598

Subjects
Secondary: 14G25: Global ground fields

#### Citation

Poonen, Bjorn. Uniform first-order definitions in finitely generated fields. Duke Mathematical Journal 138 (2007), no. 1, 1--21. doi:10.1215/S0012-7094-07-13811-0. http://projecteuclid.org/euclid.dmj/1178738559.

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