In Patrick Morandi's Book Field and Galois Theory, i was reading the Natural Irrationalities Theorem
Let $K$ be a finite Galois extension of $F$ and let $L$ be any arbitrary extension of $F$. Then $KL/L$ is Galois and $Gal(KL/L)\cong Gal(K/K\cap L)$ and $[K:F]=[K:K\cap L]$
Now in the proof they first define the map $\theta: Gal(KL/L)\to Gal(K/F)$ by $\sigma\mapsto \sigma|_K$ Now they claim this map is well defined since $K$ is normal over $F$ and $\theta$ is a group homomorphism.
Why $K/F$ normal implies $\theta$ is well defined. $\theta $ is well defined by its definition cause for all $a\in K$ if $\sigma=\sigma'$ then $\sigma|_K(a)=\sigma(a)=\sigma(a')=\sigma'|_K(a)$. Then where $K/F$ normal is coming