I've been studying algebra. In trying an exercise, I've come across the following issue.
I know that $\mathbb{Q}(a) = \mathbb{Q}(b)$, for some $a,b \notin \mathbb{Q}$. I want to conclude that $a$ and $b$ have the same minimal polynomial over $\mathbb{Q}$.
If $p(x)$ is the minimal polynomial of $a$ and $q(x)$ is the minimal polynomial of $b$ over $\mathbb{Q}$, then $ \mathbb{Q}[x]/(p(x)) \simeq \mathbb{Q}(a) = \mathbb{Q}(b) \simeq \mathbb{Q}[x]/(q(x)). $ It's clear that $p(x)$ and $q(x)$ are irreducible, monic, and have the same degree. However, I can't find a proof or a counterexample to determine whether $p(x)$ must equal $q(x)$.
Any advice? Thanks.