  An integral domain is said to have the IDF property when every non-zero
element of it has only a finite number of non-associate irreducible divisors. A
counterexample has already been found showing that IDF property does not
necessarily ascend in polynomial extensions.
  In this paper, we introduce a new class of integral domains, called
MCD-finite domains, and show that for any domain $D$, $D[X]$ is an IDF domain
if and only if $D$ is both IDF and MCD-finite. This result entails all the
previously known sufficient conditions for the ascent of the IDF property.
  Our new characterization of polynomial domains with the IDF property enables
us to use a different construction and build another counterexample which
strengthen the previously known result on this matter.
