Pseudo-algebraically closed fields over rational function fields

Jarden, Moshe; Shelah, Saharon

doi:10.1090/S0002-9939-1983-0681825-4

Pseudo-algebraically closed fields over rational function fields
HTML articles powered by AMS MathViewer

by Moshe Jarden and Saharon Shelah PDF

Proc. Amer. Math. Soc. 87 (1983), 223-228 Request permission

Abstract:

The following theorem is proved: Let $T$ be an uncountable set of algebraically independent elements over a field ${K_0}$. Then $K = {K_0}(T)$ is a Hilbertian field but the set of $\sigma \in G(K)$ for which $\tilde K(\sigma )$ is PAC is nonmeasurable.

References

James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 229613, DOI 10.2307/1970573

A mathematical approach to some problems in number theory

Gerhard Frey, Pseudo algebraically closed fields with non-Archimedean real valuations, J. Algebra 26 (1973), 202–207. MR 325584, DOI 10.1016/0021-8693(73)90020-3
Michael Fried, Dan Haran, and Moshe Jarden, Galois stratification over Frobenius fields, Adv. in Math. 51 (1984), no. 1, 1–35. MR 728998, DOI 10.1016/0001-8708(84)90002-1
Eizi Inaba, Über den Hilbertschen Irreduzibilitätssatz, Jpn. J. Math. 19 (1944), 1–25 (German). MR 16749, DOI 10.4099/jjm1924.19.1_{1}
Moshe Jarden, Elementary statements over large algebraic fields, Trans. Amer. Math. Soc. 164 (1972), 67–91. MR 302651, DOI 10.1090/S0002-9947-1972-0302651-9
Moshe Jarden, The elementary theory of $\omega$-free Ax fields, Invent. Math. 38 (1976/77), no. 2, 187–206. MR 435051, DOI 10.1007/BF01408571
Moshe Jarden, An analogue of Čebotarev density theorem for fields of finite corank, J. Math. Kyoto Univ. 20 (1980), no. 1, 141–147. MR 564673, DOI 10.1215/kjm/1250522325
Moshe Jarden and Ursel Kiehne, The elementary theory of algebraic fields of finite corank, Invent. Math. 30 (1975), no. 3, 275–294. MR 435050, DOI 10.1007/BF01425568