PAC fields over number fields
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 371-377.

Soient K un corps de nombres et N une extension galoisienne de qui n’est pas algébriquement close. Alors N n’est pas PAC sur K.

We prove that if K is a number field and N is a Galois extension of which is not algebraically closed, then N is not PAC over K.

@article{JTNB_2006__18_2_371_0,
     author = {Jarden, Moshe},
     title = {PAC fields over number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {371--377},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {2},
     year = {2006},
     doi = {10.5802/jtnb.550},
     mrnumber = {2289430},
     zbl = {05135402},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.550/}
}
Jarden, Moshe. PAC fields over number fields. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 371-377. doi : 10.5802/jtnb.550. http://www.numdam.org/articles/10.5802/jtnb.550/

[1] M. D. Fried, M. Jarden, Field Arithmetic. Second edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2005. | MR 2102046 | Zbl 1055.12003

[2] W.-D. Geyer, M. Jarden, PSC Galois extensions of Hilbertian fields. Mathematische Nachrichten 236 (2002), 119–160. | MR 1888560 | Zbl 1007.12003

[3] G. J. Janusz, Algebraic Number Fields. Academic Press, New York, 1973. | MR 366864 | Zbl 0307.12001

[4] M. Jarden, A. Razon, Pseudo algebraically closed fields over rings. Israel Journal of Mathematics 86 (1994), 25–59. | MR 1276130 | Zbl 0802.12007

[5] M. Jarden, A. Razon, Rumely’s local global principle for algebraic P𝒮C fields over rings. Transactions of AMS 350 (1998), 55–85. | MR 1355075 | Zbl 0924.11092

[6] S. Lang, Introduction to Algebraic Geometry. Interscience Publishers, New York, 1958. | MR 100591 | Zbl 0095.15301

[7] J. Neukirch, Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper. Inventiones mathematicae 6 (1969), 296–314. | MR 244211 | Zbl 0192.40102

[8] A. Razon, Splitting of ˜/. Archiv der Mathematik 74 (2000), 263–265 | MR 1742636 | Zbl 0954.12001