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here is an example;

$K$ = $\mathbb{Q} (a)$ ($a$ is a root of $x^2-3x-30 \in \mathbb{Q}[x]$ )

$O_K$ = the ring of integers of $K$

$P$ = ($-6a-25$) (P is a prime ideal of $O_K$)

$k$ = $O_K/P$

$f(x) = x^2-ax+32 \in O_K[x]$

Is there a way in SAGE to determine the factorization of $\overline{f(x)}=f(x) \hspace{2pt} modP \in k[x]$.

here is an example;

K. = NumberField( x^3+3*x^2+1 )

O = K.ring_of_integers()

P = K.ideal(?+1)

Is there an easy way in sage to determine the factorization of a polynomial $f(x) \in O[x]$ , over the residue class ring $O/P$.

For example, $f(x)=x^3+3x^2+?x+1$...