Unique factorization in generalized power series rings

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.