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I was doing a question where I have to prove that $Z[\sqrt{-5}]$ is not a UFD.

There I was proving that factors of $9=(2+\sqrt{-5})(2-\sqrt{-5})$ are not reducible for that I have to define a norm such that if $f(a+b\sqrt{-5})=a^2+5b^2$

So, My question is:-Is it really necessary to define a norm?

Because a UFD can't be a ED always.

Singh Naveen
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  • The common proof uses the *multiplicativity* of the norm (not ED) - see the linked dupe. As explained there, the norm map helps us study factorization theory in algebraic integers via the "simpler" factorization of the norms. This is independent of the ED property (some EDs use the norm as the size measure, but general EDs may use measures other than norms). – Bill Dubuque Jun 29 '22 at 18:15

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