Proposition. $K_{1} / F$ and $K_{2} / F$ Galois $\implies K_{1} \cap K_{2} / F$ and $K_{1}K_{2} / F$ Galois [2406]
Proposition. $K_{1} / F$ and $K_{2} / F$ Galois $\implies K_{1} \cap K_{2} / F$ and $K_{1}K_{2} / F$ Galois [2406]
Let $K_{1}, K_{2}$ be finite Galois extensions of $F$.
- $K_{1} \cap K_{2}$ and $K_{1}K_{2}$ are Galois over $F$.
- $\mathrm{Gal}(K_{1}K_{2} / F)$ is isomorphic to$$H = \{ (\sigma, \tau) \mid \sigma |_{K_{1} \cap K_{2}} = \tau_{K_{1} \cap K_{2}} \} \leq \mathrm{Gal}(K_{1} / F) \times \mathrm{Gal}(K_{2} / F).$$In particular, if $K_{1} \cap K_{2} = F$, then $\mathrm{Gal}(K_{1}K_{2} / F) \cong \mathrm{Gal}(K_{1} / F) \times \mathrm{Gal}(K_{2} / F)$.