Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T08:23:34.775Z Has data issue: false hasContentIssue false
  • English
  • Français

PRIMITIVE RECURSIVE DECIDABILITY FOR THE RING OF INTEGERS OF THE COMPOSITUM OF ALL SYMMETRIC EXTENSIONS OF ℚ

Part of: General field theory

Published online by Cambridge University Press:  08 May 2020

MOSHE JARDEN  and
AHARON RAZON
MOSHE JARDEN
Affiliation:
Tel Aviv University, Tel Aviv, Israel, e-mail: [email protected]
AHARON RAZON
Affiliation:
Elta Systems Ltd, Ashdod, Israel, e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ℚsymm be the compositum of all symmetric extensions of ℚ, i.e., the finite Galois extensions with Galois group isomorphic to Sn for some positive integer n, and let ℤsymm be the ring of integers inside ℚsymm. Then, TH(ℤsymm) is primitive recursively decidable.

MSC classification

Primary: 12E30: Field arithmetic
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 291 - 296
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

In memory of Wulf-Dieter Geyer (1939–2019)

References

Darnière, L., Decidability and local-global principles, in Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Denef, J., Lipshitz, L., Pheidas, T. and Geel, J. V., Editors), Contemporary Mathematics, vol. 270 (AMS, Providence, 2000), 145–167.CrossRefGoogle Scholar
van den Dries, L., Elimination theory for the ring of algebraic integers, J. Reine Angew. Math. 388 (1988), 189–205.Google Scholar
Fried, M. and Jarden, M., Field Arithmetic, 3rd edition, Ergebnisse der Mathematik, vol. 11 (Springer, Heidelberg, 2008).Google Scholar
Geyer, W.-D., Jarden, M. and Razon, A., On stabilizers of algebraic function fields of one variable, Adv. Geom. 17(2) (2017), 131–174.CrossRefGoogle Scholar
Geyer, W.-D., Jarden, M. and Razon, A., Strong approximation theorem for absolutely integral varieties over ${\rm{P}}{\cal S}{\rm{C}}$ Galois extensions of global fields, N.Y. J. Math. 23 (2017), 1447–1529.Google Scholar
Geyer, W.-D., Jarden, M. and Razon, A., Composita of symmetric extensions of ℚ, Münster J. Math. 12 (2019), 139–161.Google Scholar
Jarden, M., Algebraic Patching, Springer Monographs in Mathematics (Springer, Heidelberg, 2011).Google Scholar
Jarden, M. and Razon, A., Pseudo algebraically closed fields over rings, Isr. J. Math. 86 (1994), 25–59.CrossRefGoogle Scholar
Jarden, M. and Razon, A., Strong approximation theorem for absolutely integral varieties over the compositum of all symmetric extensions of a global field, Glasgow Math. J. 61 (2018), 373–380.CrossRefGoogle Scholar
Jarden, M. and Shlapentokh, A., Decidable algebraic fields, J. Symbolic Logic 82 (2017), 474–488.CrossRefGoogle Scholar
Moret-Bailly, L., Groupes de Picard et problémes de Skolem II, Annales Scientiques de l’Ecole Normale Superieure 22(4) (1989), 181–194.CrossRefGoogle Scholar
Neumann, K., Every finitely generated regular field extension has a stable transcendence base, Isr. J. Math. 104 (1998), 221–260.CrossRefGoogle Scholar
Razon, A., Primitive recursive decidability for large rings of algebraic integers, Albanian J. Math. 13(1) (2019), 3–93.Google Scholar
Weispfenning, V., Quantifier elimination and decision procedure for valued fields, in Models and Sets, LNM, vol. 1103 (Springer Verlag, Heidelberg, 1984), 419–472.CrossRefGoogle Scholar