Some of the proofs of this claim involves showing that 47 cannot be a norm of any element in $\mathbb{Z}[e^{2i\pi/23}]$. I see that this implies that any ideal of norm 47 cannot be principal, but how do we know that there are indeed ideals of norm 47. For cyclotomic fields I learned the following as the norm function:
$N(A) = \prod_{n=0}^{p - 1} \sigma_{n}(A)$, where $\sigma_{n}: \mathbb{Q}[e^{2i\pi/p}] \to \mathbb{Q}[e^{2i\pi/p}]$ is a homomorphism defined by $e^{2i\pi/p} \to (e^{2i\pi/p})^n$ and A is an ideal.
What are some examples of ideals A with norm 47?
