Let $E/F$ be a finite Galois extension. Let $K$ be a function field of transcendence degree one over $F$. Let $KE$ be the compositum of $K$ and $E$. Why is $KE/K$ also finite and Galois?
Also, why is $[KE:K]\leq [E:F]$?
Let $E/F$ be a finite Galois extension. Let $K$ be a function field of transcendence degree one over $F$. Let $KE$ be the compositum of $K$ and $E$. Why is $KE/K$ also finite and Galois?
Also, why is $[KE:K]\leq [E:F]$?