Simultaneous irreducibility of minimal polynomials 
Let $F$ be a field. Let $u,v$ be elements in an algebraic extension of $F$ with minimal polynomials $f$ and $g$ respectively. Prove that $g$ is irreducible over $F(u)$ if and only if $f$ is irreducible over $F(v)$. 

I have only obtained that $f,g$ are irreducible over $F$. 

$g$ is irreducible over $F(u)$ if and only if $[F(u,v):F(u)]=\deg g$ and $f$ is irreducible over $F(v)$ if and only if $[F(u,v):F(v)]=\deg f$. If consider $[F(u,v):F]$ you are done. 
