TITLE: How would one describe set of elements of $\mathbb{Z}(\sqrt{-d})$ whose norm are divisible by a rational prime $p$?
QUESTION [6 upvotes]: If we consider non-UFD's with $d > 1$, how would one go about describing the integers in $\mathbb{Z}(\sqrt{-d})$ whose norm is divisible by a prime $p$ but under the stipulation that it must be done using principal Ideals? Can it be done? I'll detail what I mean;
An example for a UFD would be the following;
Let $f(\alpha)=1$ for $\alpha \in \mathbb{Z}(\sqrt{-1})$, and 0 otherwise.. I want a new function, $g(\alpha)$, such that $g(\alpha)=1$ if $p \mid N\alpha$, otherwise $g(\alpha)=0$. For an inert prime $p$, $g(\alpha)=f(\frac{\alpha}{p})$. If p splits, then $g(\alpha)=f(\frac{\alpha}{\mathfrak{p}})+f(\frac{\alpha}{\bar{\mathfrak{p}}})-f(\frac{\alpha}{N(\mathfrak{p})})$ where $N(\mathfrak{p})=p$. In other words, how can we construct $g$ using only $f$ for Imaginary quadratic fields without unique factorization? An example would be much appreciated.

REPLY [4 votes]: The functions $f$ and $g$ can be defined, but they have to operate on ideals rather than numbers, and those ideals may or may not be principal.
First, though, let's tidy up the notation. $f_R(\alpha) = 1$ (or True) if $\alpha$ is a number in the ring $R$, and $0$ (or False) otherwise. And $g_R^p(\alpha) = 1$ if $\alpha$ is a number in the prime ideal generated by $p$, or by a splitting factor of $p$, or co-generated by $p$.
Now let's examine $\alpha = 82$, $p = 41$ and $R = \textbf{Z}[i]$. Since $41$ splits as $(4 - 5i)(4 + 5i)$, we have $$g_R^p(\alpha) = f_{\textbf{Z}[i]}\left(\frac{82}{4 - 5i}\right) + f_{\textbf{Z}[i]}\left(\frac{82}{4 + 5i}\right) - f_{\textbf{Z}[i]}\left(\frac{82}{41}\right) = 1 + 1 - 1 = 1.$$ That's correct, right?
Now try $\alpha = 86$, $p = 43$. As $43$ is inert, we have $g_R^p(\alpha) = 1$, as expected.
Let's move on to $g_{\textbf{Z}[\sqrt{-5}]}^{41}(82)$. Since $41$ splits as $(6 - \sqrt{-5})(6 + \sqrt{-5})$, the result should be much the same as $g_{\textbf{Z}[i]}^{41}(82)$.
And now we get to something really interesting: $g_{\textbf{Z}[\sqrt{-5}]}^{43}(76 + 18 \sqrt{-5})$. $43$ is inert in this ring, and yet $N(76 + 18 \sqrt{-5}) = 7396$, a multiple of $43$. Worse, $$\frac{76 + 18 \sqrt{-5}}{43} = \frac{2}{43}(38 + 9 \sqrt{-5}),$$ so $$f_{\textbf{Z}[\sqrt{-5}]}\left(\frac{76 + 18 \sqrt{-5}}{43}\right) = 0.$$
If instead we defined $g_R^p(\alpha)$ to look for the containment of $\langle \alpha \rangle$ in $\langle p \rangle$ or $\langle p, x \rangle$ or $\langle p, \overline x \rangle$, we might get the result we want, since $\langle 76 + 18 \sqrt{-5} \rangle$ is in either $\langle 43, 9 + \sqrt{-5} \rangle$ or $\langle 43, 9 - \sqrt{-5} \rangle$, one of those.