Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T19:54:55.455Z Has data issue: false hasContentIssue false
  • English
  • Français

The Absolute Galois Group of the Field of Totally S-Adic Numbers

Published online by Cambridge University Press:  11 January 2016

Dan Haran ,
Moshe Jarden  and
Florian Pop
Dan Haran
Affiliation:
School of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel, haran@post.tau.ac.il
Moshe Jarden
Affiliation:
School of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel, jarden@post.tau.ac.il
Florian Pop
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA, pop@math.upenn.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a finite set S of primes of a number field K and for σ1,…, σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1,…,σe in Ktot,S by Ktot,S(σ). We prove that foralmost all σ ∈ Gal(K)e the absolute Galois group of Ktot,S(σ) is the free product of and a free product of local factors over S.

Keywords

12E30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 194 , 2009 , pp. 91 - 147
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[BSJ] Bary-Soroker, L. and Jarden, M., PAC fields over finitely generated fields, Mathematische Zeitschrift, 260 (2008), 329–334.CrossRefGoogle Scholar
[Efr] Efrat, I., Lifting of Generating Subgroups, Proceedings of the American Mathematical Society, 125 (1997), 2217–2219.Google Scholar
[FHV] Fried, M., Haran, D., and Völklein, H., Absolute Galois group of the totally real numbers, C. R. Acad. Sci. Paris, 317 (1993), 995–999.Google Scholar
[FrJ] Fried, M. D. and Jarden, M., Field Arithmetic, Third Edition, revised by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2008.Google Scholar
[Gey] Geyer, W.-D., Galois groups of intersections of local fields, Israel Journal of Mathematics, 30 (1978), 382–396.CrossRefGoogle Scholar
[GeJ] Geyer, W.-D. and Jarden, M., PSC Galois extensions of Hilbertian fields, Mathe-matische Nachrichten, 236 (2002), 119–160.Google Scholar
[GPR] Green, B., Pop, F., and Roquette, P., On Rumely’s local-global principle, Jahres-bericht der Deutschen Mathematiker-Vereinigung, 97 (1995), 43–74.Google Scholar
[Har] Haran, D., On closed subgroups of free products of profinite groups, Proceedings of the London Mathematical Society, 55 (1987), 266–289.CrossRefGoogle Scholar
[HaJ1] Haran, D. and Jarden, M., The absolute Galois group of a pseudo real closed algebraic field, Pacific Journal of Mathematics, 123 (1986), 55–69.Google Scholar
[HaJ2] Haran, D. and Jarden, M., The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik, 383 (1988), 147–206.Google Scholar
[HaJ3] Haran, D. and Jarden, M., Regular split embedding problems over complete valued fields, Forum Mathematicum, 10 (1998), 329–351.Google Scholar
[HJPa] Haran, D., Jarden, M., and Pop, F., Projective group structures as absolute Galois structures with block approximation, Memoirs of AMS, 189 (2007), 1–56.Google Scholar
[HJPb] Haran, D., Jarden, M., and Pop, F., P-adically projective groups as absolute Galois groups, International Mathematics Research Notices, 32 (2005), 1957–1995.Google Scholar
[Jar1] Jarden, M., Algebraic realization of p-adically projective groups, Compositio Mathematica, 79 (1991), 21–62.Google Scholar
[Jar2] Jarden, M., 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.Google Scholar
[Jar3] Jarden, M., Large normal extensions of Hilbertian fields, Mathematische Zeitschrift, 224 (1997), 555–565.Google Scholar
[Jar4] Jarden, M., PAC fields over number fields, Journal de Théorie des Nombres de Bordeaux, 18 (2006), 371–377.Google Scholar
[JaR1] Jarden, M. and Razon, A., Pseudo algebraically closed fields over rings, Israel Journal of Mathematics, 86 (1994), 25–59.Google Scholar
[JaR2] Jarden, M. and Razon, A., Rumely’s local global principle for algebraic PSC fields over rings, Transactions of AMS, 350 (1998), 55–85.Google Scholar
[Lan] Lang, S., Algebra, Third Edition, Eddison-Wesley, Reading, 1993.Google Scholar
[Mel] Melnikov, O. V., Subgroups and homology of free products of profinite groups, Math. USSR Izvestiya, 34 (1990), 97–119.CrossRefGoogle Scholar
[Pop1] Pop, F., Galoissche Kennzeichnung p-adisch abgeschlossener Körper, Journal für die reine und angewandte Mathematik, 392 (1988), 145–175.Google Scholar
[Pop2] Pop, F., Fields of totally Σ-adic numbers, manuscript, Heidelberg, 1992 Google Scholar
[Pop3] Pop, F., On prosolvable subgroups of profinite free products and some applications, manuscripta mathematica, 86 (1995), 125–135.CrossRefGoogle Scholar
[Pop4] Pop, F., Embedding problems over large fields, Annals of Mathematics, 144 (1996), 1–34.Google Scholar
[Pre] Prestel, A., Lectures on Formally Real Fields, Lecture Notes in Mathematics 1093, Springer, Berlin, 1984.Google Scholar
[PrR] Prestel, A. and Roquette, P., Formally p-adic Fields, Lecture Notes in Mathematics 1050, Springer, Berlin, 1984.Google Scholar