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PAC fields over number fields. (English) Zbl 1146.12003
The aim of the paper is to prove the following theorem: Let $$N$$ be a Galois extension of a number field $$K$$ which is different from $$\overline{\mathbb Q}$$. Then $$N$$ is not PAC over $$K$$.
The main ingredient of the proof are: a result of Radon about fields which are PAC over subfields, Frobenius density theorem and J. Neukirch’s paper [“Kennzeichnung der $$p$$-adischen und der endlichen algebraischen Zahlkörper”, Invent. Math. 6, 296–314 (1969; Zbl 0192.40102)].

##### MSC:
 1.2e+31 Field arithmetic
##### Keywords:
field arithmetic; PAC fields
Full Text:
##### References:
 [1] M. D. Fried, M. Jarden, Field Arithmetic. Second edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2005. · Zbl 1055.12003 [2] W.-D. Geyer, M. Jarden, PSC Galois extensions of Hilbertian fields. Mathematische Nachrichten 236 (2002), 119-160. · Zbl 1007.12003 [3] G. J. Janusz, Algebraic Number Fields. Academic Press, New York, 1973. · Zbl 0307.12001 [4] M. Jarden, A. Razon, Pseudo algebraically closed fields over rings. Israel Journal of Mathematics 86 (1994), 25-59. · Zbl 0802.12007 [5] M. Jarden, A. Razon, Rumely’s local global principle for algebraic P $$\mathcal{S}$$ C fields over rings. Transactions of AMS 350 (1998), 55-85. · Zbl 0924.11092 [6] S. Lang, Introduction to Algebraic Geometry. Interscience Publishers, New York, 1958. · Zbl 0095.15301 [7] J. Neukirch, Kennzeichnung der $$p$$-adischen und der endlichen algebraischen Zahlkörper. Inventiones mathematicae 6 (1969), 296-314. · Zbl 0192.40102 [8] A. Razon, Splitting of $$\tilde{\mathbb{Q}}/\mathbb{Q}$$. Archiv der Mathematik 74 (2000), 263-265 · Zbl 0954.12001
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