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I need help understanding what the final sentence below means. Paraphrased from Lang's Algebra, Chapter VI, proof of Theorem 1.2 (page 266, third edition):

Let $K/k$ be Galois and $F$ an arbitrary extension, $K,F$ subfields of some other field. Then $KF/F$ and $K/K\cap F$ are Galois. Let $H=\mathrm{Gal}(KF/F)$, $G=\mathrm{Gal}(K/k)$. If $\sigma \in H$ then the restriction of $\sigma$ to $K$ is in $G$, and the map $\sigma \mapsto \sigma |_K$ gives an injective map of $H$ into the Galois group of $K/K \cap F$. Let $H'$ denote the image of this map.

It's the "and the map..." part I'm not sure about. Reasonably, the map $\sigma \mapsto \sigma|_K$ is from $H$, but into what? I'm guessing into $H$. In the proof it is assumed to be obvious that $\sigma|_K$ fixes $K \cap F$. If $H'$ is the image of $\sigma \mapsto \sigma|_K$ ($H \rightarrow H$) and $\sigma \in H'$, then $\sigma$ is supposed to leave $F$ unchanged. Since $K\cap F \subset F$, that field is also left unchanged. But how does that render the group $H'$ to be $\mathrm{Gal}(K/K\cap F)$? This group does not leave all of $F$ unchanged; nothing is said about $F\setminus (K\cap F)$. I don't get this...

Anyway, the result and its proof are not the issue, but what the codomain (target set?) of $\sigma \mapsto \sigma|_K$ is, and in particular what can be said about the order of $\mathrm{Gal}(K/K\cap F)$ if we know the orders of $G,H$.

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It's a map into $G=\mathrm{Gal}(K/k)$ - this is hinted at by the statement

If $\sigma \in H$ then the restriction of $\sigma$ to $K$ is in $G$

An element $\sigma\in H=\mathrm{Gal}(KF/F)$ is an isomorphism $$\sigma\colon KF\to KF$$ such that $\sigma(x)=x$ for all $x\in F$. The restriction of $\sigma$ to $K$ will be an isomorphism $\sigma|_K\colon K\to K$, which is not an isomorphism from $KF$ to $KF$, and therefore can't possibly be interpreted as an element of $H$. It is however an element of $\mathrm{Gal}(K/k)$ because $\sigma|_K(x)=x$ for all $x\in k$.

Note that it makes no sense to ask whether or not an element of $\mathrm{Gal}(K/K\cap F)$ "fixes the rest of $F$" or not, because elements of this group are functions that are not defined on parts of $F$ other than $K\cap F$. For example, does the automorphism $\psi:\mathbb{Q}(\sqrt{2})\to\mathbb{Q}(\sqrt{2})$ that sends $\sqrt{2}$ to $-\sqrt{2}$ "fix" $\sqrt{3}$ or not?

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  • $\begingroup$ Alright, makes sense, mostly ;) I'll return to it later, brain's too mushy at 7am. Thanks! $\endgroup$ – Erik Vesterlund Jul 9 '13 at 4:59

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