Unique factorization domains

Asked Jul 21 '20 at 18:43

Active Jul 21 '20 at 18:47

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I am confused about showing the ring $\mathbb Z[\sqrt{2}] $ is a UFD?

I understand to show that for all non-zero units you must show they factor into irreducible elements and that they must all be associates of each other although I do not know how to prove this because to disprove it is easy simply showing a counterexample.. Any help would be greatly appreciated

edited Jul 21 '20 at 18:47

Arctic Char

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asked Jul 21 '20 at 18:43

user2111154

Welcome to Mathematics Stack Exchange. In fact it's norm-Euclidean – J. W. Tanner Jul 21 '20 at 18:48
Show that $\mathbb{Z}[\sqrt{2}]$ has class number 1, which is easy using the Minkowski's bound. – user10354138 Jul 21 '20 at 18:56
3

Or you can show it is a Euclidean domain for the norm as a euclidean function. – Bernard Jul 21 '20 at 19:01
How would you show it is a Euclidean Domain? – user2111154 Jul 21 '20 at 19:07

Unique factorization domains

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