Let $F$ be a field of characteristic zero, and let $a$ be algebraic over $F$, so $K=F(a)$ is a finite separable field extension. Assume that $b$ is also a generator for $K$ over $F$, namely $K=F(b)$. Also assume that $a+b \notin F$. (Notice that the degree of the minimal polynomial of $a$ over $F$ equals the degree of the minimal polynomial of $b$ over $F$).
Is it possible to tell when $a+b$ is also a generator for $K$ over $F$? For example, if $[K:F]$ is a prime number, then $F(a+b)=K$.
See also this question which reminds that for all but finitely many $F \ni \lambda$'s, $F(a+\lambda b)=F(a)$ (in our case $F(a)=F(b)=F(a,b)$); however, it seems that there is no way to tell if $\lambda=1$ will work unless additional details are given.