Given $\zeta_{23} = e^{\frac{2\pi i}{23}}$, the extension $\mathbb{Z}[\zeta_{23}]$ is not a unique factorization domain according to many sources.
Because I don't know how to work with these things (e.g., in Sage, attempting to factor in this ring creates an infinite loop), I am interested in an example of an element of $\mathbb{Z}[\zeta_{23}]$ that factors into multiple distinct products of irreducibles. If possible, we would like to use a prime $p \in \mathbb{Z}$ that is reducible in $\mathbb{Z}[\zeta_{23}]$ for the example, and document your procedure for producing the example such that an outsider can follow.
