If $E$ and $F$ are field, what is $E\vee F$? If $E$ and $F$ are field, what is $E\vee F$ ? Let take for exemple $E=\mathbb Q(a)$ and $\mathbb Q(b)$. Do we have $$E\vee F=\mathbb Q(a,b) \ \ ?$$ 

Aren't $E$ and $F$ subfields of a same field $k$? Then $E \vee_k F$ would be the intersection of all subfields of $k$ containing both $E$ and $F$.
Else, I am not sure there is a smallest field containing any two fields of same characteristic, nor am I sure there always is a field containing them.
Maybe others can correct me.
