NECESSARY AND SUFFICIENT CONDITIONS FOR UNIQUE FACTORIZATION IN ℤ[(-1 + √<i>d</i>)/2]

Let d be an integer, α = (-1 + √d) /2 if d ≡ 1 (mod 4), and α = √d otherwise. In this note we present elementary necessary and sufficient conditions for ℤ[α] to be a unique factorization domain. We then apply this result to produce sufficient conditions for ℤ[α] to be a unique factorization domain, in terms of prime-producing quadratic polynomials. We also apply this criterion to give an improvement of Rabinowitsch's result that provides necessary and sufficient conditions for the imaginary quadratic field K = ℚ(√1-4m), m ∈ ℕ, to have class number one. We also give two non-trivial applications to real quadratic number fields.