If $F$ is a field and $x,y$ are in an algebraic extension of $F$, I'm curious as to what we can say about $[F(x+y):F]$. I can easily prove the following:

  $[F(x+y):F] \mid [F(x,y):F]$

  $[F(x+y):F] \ge \max([F(x,y):F(x)],[F(x,y):F(y)])$

  $[F(x+y):F]^2 \ge \dfrac{[F(x,y):F]}{\gcd([F(x):F],[F(y):F])}$

But can more be said? And are there any simple conditions that ensure that $F(x+y) = F(x,y)$? For example, if $\gcd([F(x):F],[F(y):F]) = 1$, is it always the case? I doubt, but don't know how to construct a counter-example.