I am trying to find all the ideals of the ring $\mathbb Q[X]$.
If $I$ is a non trivial ideal of $\mathbb Q[X]$, then there exists $p(x) \in \mathbb Q[X]$. Since $I$ is an ideal and a group under addition, we have that $-p(x) \in \mathbb Q[X]$ and $np(x) \in \mathbb Q[X]$ for all $n \in \mathbb Z$. I got stuck at this point, I would appreciate suggestions to find all the ideals of this ring.
Abstract hint: $\mathbb{K}$ field $\Rightarrow \mathbb{K}[x] $ is a PID. Concrete hint: $\mathbb{Q}[x]$ is ED, so you can divide and you can look for a minimal polynomial.