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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.

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i think d is not correct.but not sure about the others –  poton Aug 7 '12 at 3:58

1 Answer 1

up vote 1 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 $(1+\sqrt{-5})(1-\sqrt{-5})$.

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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

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