Exhibit all the ideals in the ring $\frac{F[x]}{(p(x))}$, where $F$ is a field and $p(x)$ is a polynomial in $F[x]$ (describe them in terms of the factorization of $p(x)$).
Let $p={p_1}^{k_1} {p_2}^{k_2} ... {p_n}^{k_n}$ where $p_i\in F[x]$ is irreducible.
The elements of $\frac{F[x]}{(p(x))}$ are of the form $(p(x))+q$ where $q={q_1}^{l_1} {q_2}^{l_2} ... {q_m}^{l_m}$ and if $p_i$ and $q_j$ are associates then $k_i>l_j$.
I tried Chinese remainder theorem but the ideals generated by irreducibles are not comaximal.
Please give a hint. Please do not give solution.
Thanks!