A Factorization Formula for Class Number Two

Abstract

Let R be an atomic integral domain. R is a half-factorial domain (HFD) if whenever x1…xn= y1…ym for x1, …, xn, y1, …, ym irreducibles of R, then n=m. A well known result of L. Carlitz (1960, Proc. Amer. Math. Soc.11, 391–392) states that the ring of integers in a finite extension of the rationals is a HFD if and only if the class number of R is less than or equal to 2. If R is such a ring of integers with class number 2, then we use some simple Krull monoids to develop a formula for counting the number of different factorizations of any integer x into products of irreducible elements of R.