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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Elementary statements over large algebraic fields

Author: Moshe Jarden
Journal: Trans. Amer. Math. Soc. 164 (1972), 67-91
MSC: Primary 12L05; Secondary 10B99, 10N05
MathSciNet review: 0302651
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Abstract: We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all $({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e}$ the fixed field of $\{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e})$, has the following property: For any non-void absolutely irreducible variety V defined over ${k_s}({\sigma _1}, \ldots ,{\sigma _e})$ the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of $\sigma \in \mathcal {G}(\tilde Q/Q)$ (Q is the field of rational numbers) for which E holds in $\tilde Q(\sigma )$ is equal to the Dirichlet density of the set of primes p for which E holds in the field ${F_p}$ of p elements.

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Keywords: Hilbertian fields, global fields, residue fields of global fields, ultra-products of the above residue fields, the fixed fields <!– MATH ${k_s}({\sigma _1}, \ldots ,{\sigma _e})$ –> <IMG WIDTH="125" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${k_s}({\sigma _1}, \ldots ,{\sigma _e})$"> of <!– MATH $({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e}$ –> <IMG WIDTH="207" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e}$">, Krull topology, Haar measure, absolutely irreducible varieties, Hilbert irreducibility theorem, elementary statement, Dirichlet density, Riemann hypothesis for curves
Article copyright: © Copyright 1972 American Mathematical Society