Pseudo-algebraically closed fields over rational function fields

Authors:
Moshe Jarden and Saharon Shelah

Journal:
Proc. Amer. Math. Soc. **87** (1983), 223-228

MSC:
Primary 12F20; Secondary 12F99

DOI:
https://doi.org/10.1090/S0002-9939-1983-0681825-4

MathSciNet review:
681825

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Abstract | References | Similar Articles | Additional Information

Abstract: The following theorem is proved: Let be an uncountable set of algebraically independent elements over a field . Then is a Hilbertian field but the set of for which is PAC is nonmeasurable.

**[1]**J. Ax,*The elementary theory of finite fields*, Ann. of Math.**88**(1968), 239-271. MR**0229613 (37:5187)****[2]**-,*A mathematical approach to some problems in number theory*, 1969 Number Theory Institute, Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc., Providence, R.I., 1971, pp. 161-190.**[3]**G. Frey,*Pseudo-algebraically closed fields with nonarchimedean real valuations*, J. Algebra**26**(1973), 202-207. MR**0325584 (48:3931)****[4]**M. Fried, D. Haran and M. Jarden,*Galois stratification over Frobenius fields*, Advances in Math. (to appear). MR**728998 (86c:12007)****[5]**E. Inaba,*Über den Hilbertschen Irreduzibilitässatz*, Japan. J. Math.**19**(1944), 1-25. MR**0016749 (8:62a)****[6]**M. Jarden,*Elementary statements over large algebraic fields*, Trans. Amer. Math. Soc.**164**(1972), 67-97. MR**0302651 (46:1795)****[7]**-,*The elementary theory of**-free ak-fields*, Invent. Math.**38**(1976), 187-206. MR**0435051 (55:8013)****[8]**-,*An analogue of Čebotarev density theorem for fields of finite corank*, J. Math. Kyoto Univ.**20**(1980), 141-147. MR**564673 (81d:12010)****[9]**M. Jarden and U. Kiehne,*The elementary theory of algebraic fields of finite corank*, Invent. Math.**30**(1975), 275-294. MR**0435050 (55:8012)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0681825-4

Keywords:
PAC fields,
Hilbertian fields,
Haar measure

Article copyright:
© Copyright 1983
American Mathematical Society