Let $K$ be a field of characteristic $0$, $\alpha,\beta$ algebraic elements in an algebraic closure of $K$.
Let $K(\alpha,\beta)/K$ be the associated field extension. If both $K(\alpha)$ and $K(\beta)$ are both proper subfields of $K(\alpha,\beta)$, then do we have $K(\alpha,\beta)=K(\alpha+\beta)$?
I believe the answer is no, but I am struggling to find a counterexample, given that this is generically true.
By passing to a galois closure and using the fundamental theorem of galois theory, I believe this is equivalent to finding subgroups $H,K$, where $H\neq K$ of $G$ the galois group where $\alpha$ has stabiliser $H$, $\beta$ has stabiliser $K$, but the stabiliser of $\alpha+\beta$ is strictly larger than $H\cap K$.
Best Answer
$\Bbb{Q}(2^{1/4}+ i 2^{1/4},2^{1/4}- i 2^{1/4})$ is larger than $\Bbb{Q}(2^{1/4}+ i 2^{1/4})\cong \Bbb{Q}((-8)^{1/4})$ and $\Bbb{Q}(2^{1/4})$