Let $p$ be an irreducible polynomial in $\Bbb Q[x]$ and degree larger than $1$. Prove that if $p$ has two roots $r$ and $s$ whose product is $1$ then the degree of $p$ is even.
My approach: Let $Q(x)=P(1/x)P(0)x^d$. Note they have two roots in common. If they are the same polynomial, this is trivial, as otherwise the irreduciblility is contradicted. Then there are two minimal polynomials with roots $r$ and $1/r$, but then I’m stuck. Any hint would be appreciated!