Factorization in generalized power series
Alessandro Berarducci · Transactions of the American Mathematical Society · 1999
The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R((G≤0)) \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): ∑nt−1/n+1 \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 G=(R,+,0,≤) \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 ω \omega or of the form ωωα \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 G=R \mathbf {G}=\mathbf {R} . In the final part of the paper we study the irreducibility of series with finite support.