Consider $K=\mathbb{Q}(\alpha)$ with class number $h$, where $\alpha$ is a root of $f(x)=0$, and the set of primes and roots $p,r$ such that $f(r) \equiv 0 \bmod p$. It seems there is an equivalent of class number for ideals in $\mathcal{O}_K$ of the form $<p,\alpha-r>$.
It appears that the proportion of such ideals that are principal is $1/H$, where $H$ is an integer multiple of $h$. Moreover, all ideals of norm $p^H$ are principal (see the examples below).
Does anyone know of anything in the literature that explores this topic further?
Example: consider $f(x)=x^3+91$. This gives $h=9$. Empirically, the proportion of principal ideals of the above type is $1/18$, i.e. $H=18$. $f(r) \equiv 0 \bmod 47$ has the single root 28. $g(x)=1175x^2-24252x+27354$ has $g(28) \equiv 0 \bmod 47$, and the norm $N(g(\alpha))=47^9$, giving $N(g(\alpha)^2)=47^H$. There is no polynomial $u(x)$ with the root property of $g(x)$ but with $N(u(\alpha))=47^k$ for $k<9$.
$f(r) \equiv 0 \bmod 31$ has the three roots $7, 20, 4$. $m(x)=-84x^2-219x+844$ has $m(7) \equiv 0 \bmod 31$, $m(20) \not\equiv 0 \bmod 31$, $m(4) \not\equiv 0 \bmod 31$, and $N(m(\alpha))=31^6$, giving $N(m(\alpha)^3)=31^H$. Similarly there is no polynomial $u(x)$ with the root properties of $m(x)$ but with $N(u(\alpha))=31^k$ for $k<6$.
The above is the simplest cubic $f(x)$ with $H \not= h$ I have been able to find. A more extreme example is $f(x)=x^3+6876x+573$, which has $h=34$, $H=13056=384h$.
This is taken care of by Chebotarev's density theorem, or, more classically, by a theorem due to Dirichlet (proved later by Weber and Schering and others).
Take, as a simple example, the quadratic field $K = {\mathbb Q}(\sqrt{-23})$ with class number $3$. The primes of degree $1$ in $K$ that are principal in $K$ are exactly those that split in the Hilbert class field $H$, hence have density $\frac{1}{(H:{\mathbb Q})} = \frac16$. Half of the primes are inert in $K$; the remaining primes, which have density $\frac13$, are equally distributed among the two nonprincipal ideal classes.
If you start with a quadratic number field with class number $9$, then again the inert primes have density $\frac12$, and the rest is distributed equally among the $9$ ideal classes; in particular, the principal class contains $\frac1{18}$ of all primes.