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Factorization in generalized power series


Author: Alessandro Berarducci
Journal: Trans. Amer. Math. Soc. 352 (2000), 553-577
MSC (1991): Primary 06F25; Secondary 13A16, 03H15, 03E10, 12J25, 13A05
DOI: https://doi.org/10.1090/S0002-9947-99-02172-8
Published electronically: May 20, 1999
MathSciNet review: 1473431
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Abstract: The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group $\mathbf{G}$ is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring $\mathbf{R}((\mathbf{G}^{\leq 0}))$ consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): $\sum _n t^{-1/n}+1$. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway's series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If $ \mathbf{G}= (\mathbf{R}, +, 0, \leq)$ we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either $\omega$ or of the form $\omega^{\omega^\alpha}$ and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of
M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case $\mathbf{G}=\mathbf{R}$. In the final part of the paper we study the irreducibility of series with finite support.


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Additional Information

Alessandro Berarducci
Affiliation: Università di Pisa, Dipartimento di Matematica, Via Buonarroti 2, 56127 Pisa, Italy
Email: berardu@dm.unipi.it

DOI: https://doi.org/10.1090/S0002-9947-99-02172-8
Keywords: Generalized power series, ordered rings, surreal numbers, open induction, real closed fields, valuations, ordinal numbers
Received by editor(s): September 12, 1996
Received by editor(s) in revised form: July 22, 1997
Published electronically: May 20, 1999
Additional Notes: The results of this paper were presented at the A.S.L. meeting at S. Sebastian, July 9 - 15, 1996, and at the meeting “Model Theory of Fields”, Durham, July 22 - Aug. 1, 1996.
Article copyright: © Copyright 1999 American Mathematical Society

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