Question on relation between normal subgroups and normal extensions in Fundamental Theorem of Galois Theory. 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 $G$?
 A: 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 $\eta=\alpha^{-1}$.
A: I think you are also forgetting that for any $\sigma \in G = \operatorname{Gal}(E/F)$, we have $\sigma(a)$ being a root of $f(x)$ as well. We use your notation above. 
Choose $a \in K$ and $f(x)$ the minimal polynomial of $a = \alpha_1$ over $F$, since $E^H$ is a normal extension we can write
$$f(x) = (x-a_1) \ldots (x-a_n)$$
in $E^H[x]$. Now choose any $\sigma \in G$. Then $f(\sigma^{-1}(a)) =0$ and so $\sigma^{-1}(a) = a_i$ for some $1 \leq i \leq n$. Then for any $\eta \in H$, $\eta \sigma^{-1}(a) = \eta(a_i) = a_i$ and thus
$$\sigma\eta\sigma^{-1}(a) = \sigma(a_i) = a$$
and so $\sigma \eta \sigma^{-1} \in H$, so that $\sigma H \sigma^{-1} \subseteq H$. 
Now if we put the $\sigma's$ on the other side we get $$H \subseteq \sigma^{-1}H \sigma.$$
However because this holds for any $\sigma \in G$ we thus conclude that $$H \subseteq \tau H \tau^{-1}$$
for all $\tau \in G$ and so $H$ is normal.
