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):
∑
n
t
−
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
ω
ω
α
\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.