Let $F$ be a field and let $f(x)\in F[x] $ be a polynomial irreducible in $F[x]$. Let $K/F$ be a splitting field of $f(x)$. Is the field extension $K/F$ always simple? (i.e. $K=F(\alpha)$ for some $\alpha \in K$).
The primitive element theorem states that a field extension of finite index is simple if and only if there are only finitely many intermediate fields. In all the textbooks I read, after this statement there is a classical example of a field extension of index $p^2$ that is not simple. The example is the following:
$F :=\mathbb{F}_p(\tau^p,\omega^p)$ where $\tau$ and $\omega$ are indeterminates that are algebraically independent and $K:=\mathbb{F}_p(\tau,\omega)$. In this case $K$ can be seen as the splitting field of $(x^p - \tau^p)(x^p-\omega^p) \in F[x]$. However this is clearly not irreducible.
So, is the non irreducibility truly necessary for such an example? Back to the original question: the answer is yes for separable extensions, so we may ask to work with positive characteristic. In that case an irreducible polynomial is in the form $f(x)=g(x^{p^h})$ where $g(x)$ is irreducible and separable. The answer is still yes if $F$ is finite, but in general?