Factorization in generalized power series

The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group 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)) 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 G = (R, +, 0, ≤) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type α is either ω or of the form ω^ω^α 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. In the final part of the paper we study the irreducibility of series with finite support.