Originally Posted by

**TheEmptySet** This problem is from Dummit and Foote Abstract Algebra

Exhibit all of the ideals in the ring

$\displaystyle F[x]/(p(x))$ , where F is a field and $\displaystyle p(x)$ is a polynomial in $\displaystyle F[x]$ (describe them in terms of the factorization of p(x)))

I know that since F is a field that $\displaystyle F[x]$ is a PID and a UFD.

So since $\displaystyle p(x) \in F[x]$ $\displaystyle p(x)=p_1(x)p_2(x) \cdots p_n(x) $ and where each $\displaystyle p_i(x)$ is irreduceable and this is unique upto associates. Also since each $\displaystyle p_i(x)|p(x)$ the Ideal of $\displaystyle (p(x)) \subseteq p_i(x)$

First I am not sure if the above observations help me, If they do I don't see where to go from here. Second I am having a hard time visualzing what elements of this ring would look like. I think it would eliminate all elements of F[x] that have the same degree as any factor of $\displaystyle P_i(x)$.

Thanks

TES