You would have to prove that first. https://en.m.wikipedia.org/wiki/Euclid%27s_lemma:
"This property is the key in the proof of the fundamental theorem of arithmetic
 In general, to show that a domain is a unique factorization domain, it suffices to prove Euclid's lemma and ACCP."
(https://en.m.wikipedia.org/wiki/Ascending_chain_condition_on... looks more intimidating than Euclid's lemma to me, but may be easier to prove.)
Thanks for the feedback, and additional things to take a look at!