Let $f(x)=x^4-px^2+q \in \mathbb Q[x]$ be a polynomial with $p,q$ be distinct primes. Prove that $f$ it's irreducible over $\mathbb Q$. Prove that it's Galois group is the dihedral.
I proved the following lemma:
Let $f(x)=x^2+ax^2+b$ be an irreducible polynomial over $\mathbb Q$ such that $b(a^2-4b)$ and $b$ are not squares in $\mathbb Q$. Then the galois group of $f$ is $D_8$ (the dihedral group of eight elements).
I already proved the irreducibility. But I don't know how to prove that $q(p^2-4q)$ is not a square in $\mathbb Q$.