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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]
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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

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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