Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not an UFD.

I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are all irreducible. Is there someway to conclude that $\mathbb{Z}[\sqrt{-n}]$ is not a UFD based on this? Thanks.

share|improve this question
add comment

1 Answer

up vote 24 down vote accepted

If $n$ is even, then $2$ divides $\sqrt{-n}^2=-n$ but does not divide $\sqrt{-n}$, so $2$ is a nonprime irreducible. In a UFD, all irreducibles are prime, so this shows $\mathbb{Z}[\sqrt{-n}]$ is not a UFD.

Similarly, if $n$ is odd, then $2$ divides $(1+\sqrt{-n})(1-\sqrt{-n})=1+n$ without dividing either of the factors, so again $2$ is a nonprime irreducible.

This argument works equally well for $n=3$, but fails for $n=1,2$, and in fact $\mathbb{Z}[\sqrt{-1}]$ and $\mathbb{Z}[\sqrt{-2}]$ are UFDs.

share|improve this answer
    
Thanks very much, Chris! –  Danielle Intal Oct 9 '11 at 2:46
    
@ChrisEagle Why is $\mathbb{Z}[\sqrt{-2}]$ a UFD? I tried showing it's Euclidean using the same method as for the Gauss integers, but I found that the remainder could have norm equal to that of the divisor (and not strictly less). –  JessicaB Nov 23 '12 at 15:18
2  
@JessicaB: You're doing something wrong, then. Every point in the plane is distance at most $\sqrt{3}/2<1$ from a lattice point, so you can always get the remainder strictly smaller than the divisor. –  Chris Eagle Nov 23 '12 at 15:26
    
@ChrisEagle Quite right... Our lattice has edges 1 and $\sqrt{2}$ not $\sqrt{2}$ and $\sqrt{2}$. Oops. –  JessicaB Nov 23 '12 at 15:32
    
excuse why 2 is irreducible in ℤ[√-n] –  Knight Oct 16 '13 at 17:59
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.