I'm looking to prove the following:
Let $K$ be a field and suppose that $[K(\alpha):K]=p$ and $[K(\beta):K]=q$ for $p,q$ distinct primes. Then $K(\alpha+\beta):K$ has degree $pq$.
Of course I know that $K(\alpha+\beta)$ has either degree $1,p,q$ or $pq$ over $K$. But I do not know how to rule out the cases where the degree is $p$ or $q$. Is there another way?