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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
DOI: https://doi.org/10.1090/S0002-9947-1972-0302651-9
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.


References [Enhancements On Off] (What's this?)

  • [1] J. Ax, Solving diophantine problems modulo every prime, Ann. of Math. (2) 85 (1967), 161-183. MR 35 #126. MR 0209224 (35:126)
  • [2] -, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239-271. MR 37 #5187. MR 0229613 (37:5187)
  • [3] T. Frayne, A. C. Morel and D. S. Scott, Reduced direct products, Fund. Math. 51 (1962/63), 195-228. MR 26 #28. MR 0142459 (26:28)
  • [4] W. Kuyk, Generic approach to the Galois embedding and extension problem, J. Algebra 9 (1968), 393-407. MR 38 #2128. MR 0233807 (38:2128)
  • [5] -, Extension de corps Hilbertiens, J. Algebra 14 (1970), 112-124. MR 41 #1698. MR 0257044 (41:1698)
  • [6] S. Lang, Diophantine geometry, Interscience Tracts in Pure and Appl. Math., no. 11, Interscience, New York, 1962. MR 26 #119. MR 0142550 (26:119)
  • [7] -, Algebra, Addison-Wesley, Reading, Mass., 1965. MR 33 #5416. MR 0197234 (33:5416)
  • [8] A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Sci. Indust., no. 1041, Hermann, Paris, 1948. MR 10, 262. MR 0029522 (10:621d)
  • [9] -, Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Publ., vol. 29, Amer. Math. Soc., Providence, R. I., 1946. MR 9, 303. MR 0023093 (9:303c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0302651-9
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

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