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In Wikipedia (https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory), it is stated that:

The field $E^H$ is a normal extension of $F$ if and only if $H$ is a normal subgroup of $Gal(E/F)$.

"In this case, the restriction of the elements of $Gal(E/F)$ to $E^H$ induces an isomorphism between $Gal(E^H/F)$ and the quotient group $Gal(E/F)/H$".


My question is how exactly does the restriction induce the isomorphism?

Thanks for any help. A brief explanation (or direction to a reference) will suffice.

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Define

$$\Phi: Gal(E/F)\to Gal(E^H/F)\;,\;\;\Phi\sigma:=\sigma|_H$$

Observe that then $\;\ker\Phi=H\;$ . The above is clearly a homomorphism, but why is it an epimorphism? Because any element in $\;Gal(E^H/F)\;$ can be lifted up to an element in $\;Gal(E/F)\;$ (in fact, and depending on the element, it can be lifted in several ways).

You may want to read theorem 6.2.1 in page 5 here

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