I came across a statement as follows:
Given any positive integer $n$, there exists a field extension of $\Bbb Q$ of degree $n$
I know $[\Bbb Q:\Bbb Q]=1$,
$[\Bbb Q[\sqrt2]:\Bbb Q]=2$,
$[\Bbb Q[\sqrt[3]2]:\Bbb Q]=3$,
$[\Bbb Q[\sqrt2,\sqrt3]:\Bbb Q]=4, [\Bbb Q[\sqrt[4]2]:\Bbb Q]=4$
and $[\Bbb Q[\sqrt2,\sqrt3,\sqrt5...]:\Bbb Q]=\infty$.
Is $[\Bbb Q[\sqrt[p]2,\sqrt[q]3]:\Bbb Q]=pq$ (2,3 are randomly chosen primes; and $p$ and $q$ are primes)? Also is it valid for $p=q$?
Is it an established result? If they are not true, can someone provide a simple counter example?
My aim is to have a set of examples for field extension ready given any dimension $n$.
Edit: just found this answer, clarifying $[\Bbb Q[\sqrt[n]2]:\Bbb Q]=n$, $\forall n\in\Bbb N$; hence edited it out of question.