I'm looking for an example of the following kind:
Let $a,b\notin \Bbb{Q}$, where $a$ and $b$ satisfy the irreducible polynomials $p(x)$ and $q(x)\in\Bbb{Q}[x]$ respectively. The irreducible polynomial satisfied by $b$ in $\Bbb{Q}(a)[x]$ is not equal to $q(x)$. i.e. it is of a lower degree (preferably of a degree greater than $1$).
Thank you.
$a = \sqrt{2}, b = - \sqrt{2}$. For a less silly example, $a = \sqrt[3]{4}, b = \sqrt[3]{2}$.
Edit: Try $a = i$ and $b$ a root of $x^2 + (1 + i) x + (1 + i)$. This is irreducible by Eisenstein's criterion over $\mathbb{Z}[i]$, although I'm not sure how to show that $\mathbb{Q}(b)$ doesn't contain $\mathbb{Q}(i)$ (and it might not be true).