Frobenius cycle type needs good reduction

Given: The data and hypotheses of the statement.

Frobenius cycle type and prime splitting: Let $F\in \mathbb{Z}[T]$ be monic separable with splitting field L, and let p be a rational prime not dividing $\mathrm{Disc}(F)$. Then p is unramified in L and $\bar{F}\in {\mathbb{F}}_p[T]$ is squarefree. The degrees of its monic irreducible factors, with each distinct factor counted once, are exactly the cycle lengths of arithmetic Frobenius on the roots of F.

Dedekind--Kummer prime factorisation: Let $L\mathrm{/}K$ be a finite extension of number fields, let $\alpha \in {\mathcal{O}}_L$ with ${\mathcal{O}}_L={\mathcal{O}}_K[\alpha ]$, and let $F\in {\mathcal{O}}_K[X]$ be its monic minimal polynomial over $K$. Let $\mathfrak{p}$ be a nonzero prime ideal of ${\mathcal{O}}_K$, not dividing the index of this power order (the index is $1$ under the stated monogeneity hypothesis). If $\bar{F}={\prod }_i{\bar{g}}_i^{e_i}$ with distinct monic irreducibles over ${\mathcal{O}}_K\mathrm{/}\mathfrak{p}$, then $\mathfrak{p}{\mathcal{O}}_L={\prod }_i{\mathfrak{P}}_i^{e_i},{\mathfrak{P}}_i=(\mathfrak{p},g_i(\alpha )),f({\mathfrak{P}}_i\mathrm{/}\mathfrak{p})=\mathrm{deg}{\bar{g}}_i,$ Here $g_i\in {\mathcal{O}}_K[X]$ are any monic lifts of ${\bar{g}}_i$, and the last $f$ denotes the residue degree from def-prime-above-and-residue-degree.

Ramification is detected by the number-field discriminant: A rational prime $p$ ramifies in $K\mathrm{/}\mathbb{Q}$ if and only if $p\mid d_K$.

Integers in a quadratic field: For squarefree $d\mathrm{\ne }1$, ${\mathcal{O}}_{\mathbb{Q}(\sqrt{d})}=\mathbb{Z}[(1+\sqrt{d})\mathrm{/}2]$ if $d\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, and $\mathbb{Z}[\sqrt{d}]$ otherwise.

Power-basis and polynomial discriminants: Let $K=\mathbb{Q}(\alpha )$. If $f$ is the degree-$n$ monic minimal polynomial of $\alpha$, then $\mathrm{disc}(1,\alpha ,\dots ,{\alpha }^{n-1})=(-1{)}^{n(n-1)\mathrm{/}2}{N}_{K\mathrm{/}\mathbb{Q}}(f^{\mathrm{\prime }}(\alpha ))=\mathrm{disc}(f)\mathrm{.}$