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Let $n$ be a square free integer and alpha belong to $\mathbb{Z}[\sqrt{n}]$. If $N(\alpha)$ is a prime integer, then $\alpha$ is irreducible in $\mathbb{Z}[\sqrt{n}]$.

Here $N(\alpha)$ denotes the norm of $\alpha$.

How can I show this? I'll get back in the morning because it's super late right now!

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    Actually, $\alpha$ is prime, see [here](https://math.stackexchange.com/questions/2074491).2017-01-31

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