It really isn't obvious (I promise!), and Gowers's Answer 4 is "the real reason" for a mathematician. Alternatively viewed, it is not obvious that Euclidean dom

It really isn't obvious (I promise!), and Gowers's Answer 4 is "the real reason" for a mathematician. Alternatively viewed, it is not obvious that Euclidean domains are unique factorisation domains. (Though it is obvious that EDs are principal ideal domains, so the alternative view becomes "it is not obvious that PIDs are UFDs".)