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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Pick out the correct statements from the following list:

a. A homomorphic image of a UFD (unique factorization domain) is again a UFD.

b. The element $2 ∈ \Bbb{Z}[\sqrt{−5}]$ is irreducible in $\Bbb{Z}[\sqrt{−5}].$

c. Units of the ring $\Bbb{Z}[\sqrt{−5}]$ are the units of $\Bbb{Z}.$

d. The element $2$ is a prime element in $\Bbb{Z}[\sqrt{−5}].$

I think (d) is not correct. But I'm not sure about the others.

share|cite|improve this question
i think d is not correct.but not sure about the others – poton Aug 7 '12 at 3:58
up vote 2 down vote accepted

Hints: For (a) note that $\mathbb{Z}_4$ is a homomorphic image of $\mathbb{Z}$. For (b) and (c), use properties of norm. You are right about (d), since $2$ divides $(1+\sqrt{-5})(1-\sqrt{-5})$.

share|cite|improve this answer
and $2$ does not divide $1+\sqrt{-5}$ or $1-\sqrt{-5}$ – Belgi Aug 7 '12 at 4:48
@Belgi $\rm\,\ 2\nmid 1\pm\sqrt{-5}\color{#C00}{\ \ in\ \ \Bbb Z[\sqrt{-5}]}\ \ $ – Bill Dubuque Aug 7 '12 at 4:55
@BillDubuque - this is important, thanks for adding this! (this is what I meant but I forgot to mention) – Belgi Aug 7 '12 at 4:57

Your Answer


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.