Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T23:49:20.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of prime elements in rings of generalized power series

Published online by Cambridge University Press:  12 March 2014

Daniel Pitteloud
Daniel Pitteloud*
Affiliation:
Université de Lausanne, Institute D'Informatique, 1015 Lausanne, Switzerland, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge “ring” of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ring K((G≤0)) of series with non-positive exponents. Berarducci (see [1]) proved that K((G≤0)) does have irreducible elements, but it remained open whether the irreducibles are prime i.e., generate a prime ideal. In this paper we prove that K((G≤0)) does have prime elements if G = (ℝ, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1206 - 1216
Copyright
Copyright © Association for Symbolic Logic 2001

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.)

References

REFERENCES

[1]Berarducci, A., Factorization in generalized power series, Transactions of the American Mathematicall Society, vol. 352 (2000).Google Scholar
[2]Conway, J. H., On numbers and games, Academic Press, London, 1976.Google Scholar
[3]Ecalle, J., Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathematiques, (1992), Hermann, Paris.Google Scholar
[4]Gonshor, H., An introduction to the theory of surreal numbers, Cambridge University Press, Cambridge, 1986.CrossRefGoogle Scholar
[5]Hahn, H., Uber die nichtarchimedischen grossensysteme, S.B. Akad. Wiss. Wien. IIa, vol. 116 (1907), pp. 601–655.Google Scholar
[6]Kaplanski, I., Maximalfields with valuations, Duke Mathematical Journal, vol. 9 (1942), pp. 303–321.Google Scholar
[7]Mourgues, M. H. and Ressayre, J. P., Every real closed field has an integer part, this Journal, (1993), pp. 641–647.Google Scholar
[8]Pitteloud, D., Algebraic properties in rings of generalized power series, Annals of Pure and Applied Logic, to appear.Google Scholar
[9]Pohlers, W., Proof Theory, (Dold, A., Eckmann, B., and Takens, F., editors), Lectures Notes in Mathematics, no. 1407, Springer-Verlag, Berlin Heidelberg, 1989.CrossRefGoogle Scholar
[10]Ressayre, J. P., Integers parts of real closed exponential fields, pp. 278–288, Oxford University Press, Oxford, 1993, pp. 278–288, Arithmetic, Proof Theory and Computational Complexity (P. Clote and J. Krajicek, editors).CrossRefGoogle Scholar
[11]Ressayre, J. P., Survey on transfinite series and their applications, 1995, manuscript.Google Scholar
[12]Ribenboim, P., Fields, algebraically closed and others, Manuscript a Mathematica, vol. 75 (1992), pp. 115–166.CrossRefGoogle Scholar
[13]van den Dries, L., Macintyre, A., and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, vol. 140 (1994), pp. 183–205.CrossRefGoogle Scholar
[14]van den Dries, L., Macintyre, A., and Marker, D., Logarithmic-exponential power series, Journal of the London Mathematical Society, vol. 2 (1997), no. 56, pp. 183–205.Google Scholar
[15]van den Dries, L., Macintyre, A., and Marker, D., Logarithmic-exponential power series, preprint, 1998.CrossRefGoogle Scholar
[16]van der Hoeven, J., Asymptotique automatique, Ph.D. thesis, Université Paris 7, 1997.Google Scholar