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There is a primitive 12-th root of unity and 5 is not a prime since the minimal polynomial mod 5 is reducible. The problem is I don't know how to show 5 is irreducible or not.

What I thought was if this ring of integer is UFD then 5 is not irreducible. so I've found a source that the ring of integer is actually norm-euclidean which means that is PID so UFD.. But then the proof of the theorem seems quite beyond my course.

How to show 5 is irreducible or not?

Thanks.

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Let $\zeta$ be a primitive 12th root of unity. Then $\zeta^3$ is a primitive 4th root of unity, hence $\mathbb{Z}[\zeta]$ contains $i$.

Therefore we can write $5=(2+i)(2-i)$ in $\mathbb{Z}[\zeta]$, and by looking at their norms you can show that neither $2+i$ nor $2-i$ is a unit. Hence $5$ is not irreducible in $\mathbb{Z}[\zeta]$.

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