Consider the following proposition: Let $K/F$ be a finite Galois extension with Galois group $G$, and let $H$ be a subgroup of $G$. $K$ and $F$ here have characteristic 0. Prove that there exists an element $β ∈ K$ whose stabilizer is equal to $H$.
Here are my ideas:
(1) Since $K/F$ is finite and $char(F) = 0$, there exists a $\gamma \in K$ such that $K = F(\gamma)$. (The Primitive Elements Theorem)
(2) Consider the stabilizer of $\gamma: Stab(\gamma) = \{\sigma \in H | \sigma(\gamma) = \gamma\}$ (Is this correct definition of the stabilizer here?)
(3) Indeed $Stab(\gamma) \subset H$
(4) How do I show that $H \subset Stab(\gamma)$? Can anyone show me a way to move forward?
Consider $L=\{x\in K, h(x)=x,h\in H\}$, $L$ is a subfield of $K$. The primitive element implies that $L=F(a)$, $H\subset Stab(a)$ since $a\in L$.
Let $g:g(a)=a$, let $x\in L, x=c_0+c_1a+c_2a^2+..+c_na^n$ $c_i\in F$ since $L=F(a)$, this implies that $g(x)=g(c_0+c_1a+c_2a^2+..+c_na^n)=c_0+c_1g(a)+..+c_ng(a_n)=c_0+c_1a+c_2a^2+..+c_na^n=x$, we deduce that for every $x\in L, g(x)=x$, implies $g\in H$ and $Stab(a)=H$.