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Transactions of the American Mathematical Society
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0302651-9
PII: S 0002-9947(1972)0302651-9
Keywords: Hilbertian fields, global fields, residue fields of global fields, ultra-products of the above residue fields, the fixed fields $ {k_s}({\sigma _1}, \ldots ,{\sigma _e})$ of $ ({\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