Nonunique factorization in number rings

Question

Can you give an example of a nonunique factorization in $\mathbb{Z}[e^{2\pi i/23}]$?

Can you write two different decompositions into irreducibles in $\mathbb{Z}[e^{2\pi i/23}]$?

Answer 1

An explicit counterexample is given by $6533$, which has two different decompositions, namely $$ 6533=47\cdot 139= (1 - α + α^{21})(1 - α^2 + α^{21})\cdot \ldots \cdot(1 - α^{21} + α^{21}), $$ for a certain $\alpha\in \mathbb{Z}[\zeta_{23}]$. For details see the proof here, which is taken from Edwards book.

Another proof goes as follows: Since the ring of integers $\Bbb{Z}[e^{2\pi i/23}]$ in $\mathbb{Q}(\zeta_{23})$ is a UFD if and only if it is a PID, it is enough to see that $\Bbb{Z}[e^{2\pi i/23}]$ is not a PID. In fact, the ideal $Q = (2, \frac{1 + \sqrt{-23}}{2})$ is not principal in $\Bbb{Z}[e^{2\pi i/23}]$. This also leads to an explicit counterexample for unique factorization.