I've been stuck on this problem for a while now.

Suppose for relatively prime quadratic integers in . Show that for some unit and some quadratic integer in .

Any pointers on how to start?

Thanks.

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- October 27th 2008, 03:56 PMPn0yS0ld13rRelatively prime quadratic integers
I've been stuck on this problem for a while now.

Suppose for relatively prime quadratic integers in . Show that for some unit and some quadratic integer in .

Any pointers on how to start?

Thanks. - October 27th 2008, 06:48 PMNonCommAlg
we know that is a unit and is irreducible in the ring of integers of now use this fact that is a UFD to finish the proof.

- October 27th 2008, 06:59 PMThePerfectHacker
I had a different proof (

**NonCommAlg**beat me (Angry)) that did not involve finding factorizations.

Note that . Therefore, if (a non-unit) is decomposed into irreducibles it must mean that norm of each irreducible is a divisor of . Since is not a prime and it means all irreducibles must have norm . Note that is the only irreducible of norm up to associates. This immediately forces to be a unit for otherwise it would mean are not relatively prime. Thus, for some unit . - October 27th 2008, 11:11 PMPn0yS0ld13r
Thank you

**NonCommAlg**and**ThePerfectHacker**! I got it now. :)