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Proceedings of the American Mathematical Society

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Unique factorization in generalized power series rings


Authors: James Pommersheim and Shahriar Shahriari
Journal: Proc. Amer. Math. Soc. 134 (2006), 1277-1287
MSC (2000): Primary 06F25; Secondary 13A16, 03H15, 03E10, 12J25, 13A05
Published electronically: October 18, 2005
MathSciNet review: 2199170
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Abstract: Let $ K$ be a field of characteristic zero and let $ K((\mathbb{R}^{\leq 0}))$ denote the ring of generalized power series (i.e., formal sums with well-ordered support) with coefficients in $ K$, and non-positive real exponents. Berarducci (2000) constructed an irreducible omnific integer, in the sense of Conway (2001), by first proving that an element of $ K((\mathbb{R}^{\leq 0}))$ that is not divisible by a monomial and whose support has order type $ \omega$ (or $ \omega^{\omega^\alpha}$ for some ordinal $ \alpha$) must be irreducible. In this paper, we consider elements of $ K((\mathbb{R}^{\leq 0}))$ with support of order type $ \omega^2$. The irreducibility of these elements cannot be deduced solely from the order type of their support and, after developing new tools for studying these elements, we exhibit both reducible and irreducible elements of this type. We further prove that all elements whose support has order type $ \omega^2$ and which are not divisible by a monomial factor uniquely into irreducibles. This provides, in the ring $ K((\mathbb{R}^{\leq 0}))$, a class of reducible elements for which we have unique factorization into irreducibles.


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Additional Information

James Pommersheim
Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202
Email: jamie@reed.edu

Shahriar Shahriari
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
Email: sshahriari@pomona.edu

DOI: https://doi.org/10.1090/S0002-9939-05-08162-1
Keywords: Generalized power series, unique factorization, surreal numbers, omnific integers, ordered rings
Received by editor(s): January 7, 2004
Received by editor(s) in revised form: December 25, 2004
Published electronically: October 18, 2005
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society