×

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:

12E30 Field arithmetic

Citations:

Zbl 0192.40102
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML Link

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.