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PAC fields over finitely generated fields. (English) Zbl 1146.12002
The main theorem of the paper is the following: Let \(K\) be a field finitely generated over its prime field, and let \(M\) be a Galois extension of \(K\). If \(M\) is not separably closed, then \(M\) is not PAC over \(K\).
The theorem generalizes a previous result of M. Jarden [“PAC fields over number fields”, J. Théor. Nombres Bordx. 18, No. 2, 371–377 (2006; Zbl 1146.12003)], where the same statement is proved for the case when \(K\) is a number field.

MSC:
12E30 Field arithmetic
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[1] Fried, M.D., Jarden, M.: Field Arithmetic, vol. 11, 2nd edn. Revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3). Springer, Heidelberg (2005) · Zbl 1055.12003
[2] Huppert B. (1967). Endliche Gruppen I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 134. Springer, Berlin
[3] Jarden M. and Razon A. (1994). Pseudo algebraically closed fields over rings. Isr. J. Math. 86: 25–59 · Zbl 0802.12007 · doi:10.1007/BF02773673
[4] Jarden M. and Razon A. (1998). Rumely’s local global principle for algebraic \({\mathrm P}\mathcal {S}{\mathrm C}\) fields over rings Trans. AMS 350: 55–85 · Zbl 0924.11092 · doi:10.1090/S0002-9947-98-01630-4
[5] Jarden M. (2006). PAC fields over number fields. J. de Théorie des Nombres de Bordeaux 18: 371–377 · Zbl 1146.12003
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