# Decide whether z(sqrt(-5)) unique factorization domain or not (ring theory)

Decide whether $z\left(\sqrt{-5}\right)$ unique factorization domain or not (ring theory)
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aprovard
Step 1
To decide whether $Z\left(\sqrt{-5}\right)$ is unique factorization domain or not.
Step 2
Note that For a integral domain R to be unique factorization domain one of the property is:
if $a={p}_{1}\cdot {p}_{2}\cdot {p}_{3}\cdot \dots ..\cdot {p}_{n}$,
$a=q1\cdot q2\cdot q3\cdot \dots .\cdot {q}_{m}$
where p and q are irreducible in R then m=n and each ${p}_{i}$ is associative of some ${q}_{j}$.
Step 3
Here note that $46\in Z\left[\sqrt{-5}\right]$ is an non-zero and non-unit element and 46 can be expressed as:
46=2*23, and
$46=\left(1-3\sqrt{-5}\right)\cdot \left(1+3\sqrt{-5}\right)$
but 2 is not associative of $\left(1-3\sqrt{-5}\right),\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\left(1+3\sqrt{-5}\right)$
Hence, $Z\left[\sqrt{-5}\right]$ is not unique factorization domain.
Jeffrey Jordon