This is about exercise 84(ii) in Rotman "Galois theory":
Prove that any splitting field $K/F$ containing $B$ (finite extension of $F$) has the form $K=B_1 \vee... \vee B_r$ where each $B_i$ is isomorphic to $B$ via an isomorphism which fixes $F$. (Hint: If $Gal(K/F)=\{\sigma_1,...,\sigma_r \}$, then define $B_i=\sigma_i(B)$.)
Now I see that each $B_i$ as given in the hint is clearly isomoporhic to $B$ while $F$ is fixed and that $K \supseteq B_1 \vee... \vee B_r$, but I don't know how to prove the other inclusion? Any help welcome.