Q:

Irreducible polynomials with same root

If a monic polynomial $p(x)$ is irreducible over rationals and has $\alpha$ as root.  Is it possible that $\alpha$ is root of some other irreducible monic polynomial $q(x)$ such that $\deg(p(x))\neq \deg(q(x))$? 
This question arose when I was thinking about minimal polynomials of certain elements. 

A:

No if $p$ is the minimal polynomial of $\alpha$ in $\mathbb{Q}$, $p$ will divide every other polynomial which has $\alpha$ as a root.  This is because the ring of polynomials over a field is a principal ideal domain.  So the ideal $I$ of all polynomials which has a root $\alpha$ will be principal say equal to some $\langle h \rangle$, $h$ - monic.  Now if $p, q \in \langle h \rangle$ then $h$ divides $p$ and $q$ hence $h = p = q$, because $p$ and $q$ are irredicible.