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?