Power series extensions of half-factorial domains

Abstract

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R , and independent variables, Y and T , R [ [ Y ] ] [ [ T ] ] is not HFD even when R [ [ Y ] ] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.