Nagoya Mathematical Journal

The absolute Galois group of the field of totally $S$-adic numbers

Dan Haran, Moshe Jarden, and Florian Pop

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Abstract

For a finite set $S$ of primes of a number field $K$ and for $\sigma_{1}, \dots, \sigma_{e} \in \operatorname{Gal}(K)$ we denote the field of totally $S$-adic numbers by $K_{{\rm tot}, S}$ and the fixed field of $\sigma_{1}, \dots, \sigma_{e}$ in $K_{{\rm tot}, S}$ by $K_{{\rm tot}, S}({\boldsymbol\sigma})$. We prove that for almost all ${\boldsymbol\sigma} \in \operatorname{Gal}(K)^{e}$ the absolute Galois group of $K_{{\rm tot}, S}({\boldsymbol\sigma})$ is the free product of ${\hat F}_{e}$ and a free product of local factors over $S$.

Article information

Source
Nagoya Math. J., Volume 194 (2009), 91-147.

Dates
First available in Project Euclid: 17 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1245209126

Mathematical Reviews number (MathSciNet)
MR2536528

Zentralblatt MATH identifier
1261.12006

Subjects
Primary: 12E30: Field arithmetic

Citation

Haran, Dan; Jarden, Moshe; Pop, Florian. The absolute Galois group of the field of totally $S$-adic numbers. Nagoya Math. J. 194 (2009), 91--147. https://projecteuclid.org/euclid.nmj/1245209126


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References

  • L. Bary-Soroker and M. Jarden, PAC fields over finitely generated fields, Mathematische Zeitschrift, 260 (2008), 329--334.
  • I. Efrat, Lifting of Generating Subgroups, Proceedings of the American Mathematical Society, 125 (1997), 2217--2219.
  • M. Fried, D. Haran, and H. Völklein, Absolute Galois group of the totally real numbers, C. R. Acad. Sci. Paris, 317 (1993), 995--999.
  • M. D. Fried and M. Jarden, Field Arithmetic, Third Edition, revised by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2008.
  • W.-D. Geyer, Galois groups of intersections of local fields, Israel Journal of Mathematics, 30 (1978), 382--396.
  • W.-D. Geyer and M. Jarden, $PSC$ Galois extensions of Hilbertian fields, Mathematische Nachrichten, 236 (2002), 119--160.
  • B. Green, F. Pop, and P. Roquette, On Rumely's local-global principle, Jahresbericht der Deutschen Mathematiker-Vereinigung, 97 (1995), 43--74.
  • D. Haran, On closed subgroups of free products of profinite groups, Proceedings of the London Mathematical Society, 55 (1987), 266--289.
  • D. Haran and M. Jarden, The absolute Galois group of a pseudo real closed algebraic field, Pacific Journal of Mathematics, 123 (1986), 55--69.
  • D. Haran and M. Jarden, The absolute Galois group of a pseudo $p$-adically closed field, Journal für die reine und angewandte Mathematik, 383 (1988), 147--206.
  • D. Haran and M. Jarden, Regular split embedding problems over complete valued fields, Forum Mathematicum, 10 (1998), 329--351.
  • D. Haran, M. Jarden, and F. Pop, Projective group structures as absolute Galois structures with block approximation, Memoirs of AMS, 189 (2007), 1--56.
  • D. Haran, M. Jarden, and F. Pop, P-adically projective groups as absolute Galois groups, International Mathematics Research Notices, 32 (2005), 1957--1995.
  • M. Jarden, Algebraic realization of $p$-adically projective groups, Compositio Mathematica, 79 (1991), 21--62.
  • M. Jarden, Intersection of local algebraic extensions of a Hilbertian field (A. Barlotti et al., eds\.), NATO ASI Series C, 333, Kluwer, Dordrecht, 1991, pp. 343--405.
  • M. Jarden, Large normal extensions of Hilbertian fields, Mathematische Zeitschrift, 224 (1997), 555--565.
  • M. Jarden, PAC fields over number fields, Journal de Théorie des Nombres de Bordeaux, 18 (2006), 371--377.
  • M. Jarden and A. Razon, Pseudo algebraically closed fields over rings, Israel Journal of Mathematics, 86 (1994), 25--59.
  • M. Jarden and A. Razon, Rumely's local global principle for algebraic $P\calSC$ fields over rings, Transactions of AMS, 350 (1998), 55--85.
  • S. Lang, Algebra, Third Edition, Eddison-Wesley, Reading, 1993.
  • O. V. Melnikov, Subgroups and homology of free products of profinite groups, Math. USSR Izvestiya, 34 (1990), 97--119.
  • F. Pop, Galoissche Kennzeichnung $p$-adisch abgeschlossener Körper, Journal für die reine und angewandte Mathematik, 392 (1988), 145--175.
  • F. Pop, Fields of totally $\Sigma$-adic numbers, manuscript, Heidelberg, 1992
  • F. Pop, On prosolvable subgroups of profinite free products and some applications, manuscripta mathematica, 86 (1995), 125--135.
  • F. Pop, Embedding problems over large fields, Annals of Mathematics, 144 (1996), 1--34.
  • A. Prestel, Lectures on Formally Real Fields, Lecture Notes in Mathematics 1093, Springer, Berlin, 1984.
  • A. Prestel and P. Roquette, Formally $p$-adic Fields, Lecture Notes in Mathematics 1050, Springer, Berlin, 1984.