I'm self studying Jacobson's Basic Algebra I but I'm getting hung up on the proof of the Fundamental Theorem of Galois Theory in Jacobson's book on page 239.
Let $G=\operatorname{Gal}(E/F)$ for a field extension $E/F$. He is proving $H$ is normal in $G$ iff the fixed field $\operatorname{Inv}(H)$ is normal over $F$.
Proof: Suppose $K=\operatorname{Inv}(H)$ is normal over $F$. Let $a\in K$ with $f(x)$ the minimal polynomial over $F$. So $f(x)=(x-a_1)\cdots(x-a_n)$ where $a_1=a$ in $K[x]$. If $\eta\in G$, then $f(\eta(a))=0$ so $\eta(a)=a_i$ for some $i$. Thus $\eta(a)\in K$, so $\eta(K)\subset K$. As before, this implies $\eta H\eta^{-1}\subset H$ if $H$ is the subgroup corresponding to $K$ in the Galois pairing.
I don't follow the last sentence. Earlier Jacobson shows if $\operatorname{Inv}(H)=K$, then $\operatorname{Inv}(\eta H\eta^{-1})=\eta(K)$. He also proves that $H_1\supset H_2\iff\operatorname{Inv}(H_1)\subset\operatorname{Inv}(H_2)$. So here he has $$\operatorname{Inv}(\eta H\eta^{-1})=\eta(K)\subset K=\operatorname{Inv}(H),$$ but doesn't that imply $H\subseteq\eta H\eta^{-1}$ instead? How does he conclude $H$ is normal in $G$?
Best Answer
Let $\alpha=\eta^{-1}$. Since $H\subseteq \eta H\eta^{-1}$, we get that $\alpha H\alpha^{-1}\subseteq H$. Note that $\alpha \in G$ is arbitrary, as we can choose $\eta=\alpha^{-1}$.