Main Question: Does ZF (no axiom of choice) prove that every Principal Ideal Domain is a Unique Factorization Domain?

The proofs I've seen all use dependant choice.

Minor Questions: Does ZF + Countable Choice prove all PIDs are UFDs? Does ZF prove "If all PIDs are UFDs, then [some choice principle]"?

(If anyone knows how I could force line breaks to put the questions on their own lines, please tell me.)