I have to determine all singular primes of the quadratic ring $\mathbb{Z}[\sqrt{d}]$, where $d$ is a non-square integer, but it is not necessarily squarefree.
Say $d=p_1^{k_1}\ldots p_t^{k_t}$, with at least one $k_i=1$ (since then $d$ is not a square). Also, let $f=X^2-d$ be the monic polynomial. I know that I have to check when the determinant is equal to zero mod $p$, because for these primes $p$, we could have a problem of a multiple factor. Since $\Delta=4d$, we see that $p=2$ gives a problem, as well as all primes $p_i$ dividing $d$. From this, I'm getting a bit stuck. I think that for $f$ to be reducible, we have to look at the primes $p_i$ for which $p_i^2|d$ (Eisenstein). But when do I know for which $p$ the function $f=X^2-d$ is completely ramified?