Let be a square-free integer with . By considering the factorisation of , show that the ring is never a Unique Factorisation Domain.
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Originally Posted by Boysilver Let be a square-free integer with . By considering the factorisation of , show that the ring is never a Unique Factorisation Domain. first show that 2 is irreducible: if then but the equation has no integer solution since now, if was a UFD, then, because 2 is irreducible, it'd have to be prime. therefore, since , we'd have either or which is impossible.
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