I attempting to solve this proof for my Abstract Algebra II class.
Let $K$ be a finite normal extension of a field $F$, and let $x$ be an element of $G(K/F)$. Prove that if $x(b) = b$ for some element $b \in K$, then $x$ is an element of $G(K/F(b))$.
I attempted to prove by assuming that $x$ is not an element of $G(K/F(b))$ and show that it isn't possible. Though, I am failing to link the field $F$ and $F(b)$ and how the automorphism fixes both $F$ and $F(b)$. Any suggestions would be appreciated.
Thank you.