I have a few questions about ideals the ring of integers $\mathbb{Z}[\zeta_{n}]$ in a cyclotomic number field. Specifically, I'm trying to classify the ideals of norm 2.
I know that the Gaussian integers are a PID and for any ideal $(a)$ where $a=x+iy$ we have that $|\mathbb{Z}[i]/(a)|=N(a)=N(x+iy)=x^{2}+y^{2}$. Therefore the only ideal of norm 2 is $(1+i)$.
With a little work I was able to prove that this also holds for the Eisenstein integers with $N(x+\omega y)=x^{2}+y^{2}-xy$. This time $x^{2}+y^{2}-xy=2$ has no integer solutions, so conjecturally $\mathbb{Z}[i]$ has no ideals of norm 2.
In $\mathbb{Z}[\zeta_{5}]$ the norm can be written $N(\alpha)=\frac{1}{4}(A^{2}-5B^{2})$ for integers $A$ and $B$. $N(\alpha)=2 \Rightarrow A^{2}-5B^{2}=8 \Rightarrow A^{2} \equiv 3 \pmod 5$. But 3 is not a quadratic residue modulo 5, so there are no ideals of norm 2.
It seems like the $n$ for which $\mathbb{Z}[\zeta_{n}]$ is a Euclicean domain was a tough question and that there are 46 such $n$. For such $n$, my questions are these:
1) Is it true that $|\mathbb{Z}[\zeta_{n}]/(\alpha)|=N(\alpha)$?
2) For which $n$ does $\mathbb{Z}[\zeta_{n}]$ have ideals of norm 2? How many?