Exercise 6.1.3

Proof.

(a)

If $\phi :L_1\to L_2$ is a field isomorphism, and $\sigma \in \mathrm{Gal}(L_1∕F_1)$, then $\sigma :L_1\to L_1$, and so $\phi \circ \sigma \circ {\phi }^{-1}$ is a map from $L_2$ to $L_2$, composed of three field isomorphisms. Therefore $\phi \circ \sigma \circ {\phi }^{-1}$ is an automorphism of $L_2$.

Moreover, if $\alpha \in F_2$, then ${\phi }^{-1}(\alpha )\in F_1$, since $F_2=\phi (F_1)$. As $\sigma \in \mathrm{Gal}(L_1∕F_1)$, $\sigma$ is identity on $F_1$, thus $\sigma ({\phi }^{-1})(\alpha )={\phi }^{-1}(\alpha )$, and $(\phi \circ \sigma \circ {\phi }^{-1})(\alpha )=\alpha$. Consequently

$$\phi \circ \sigma \circ {\phi }^{-1}\in \mathrm{Gal}(L_2∕F_2).$$

Let

$$\chi :\{\begin{array}{ccc}\hfill \mathrm{Gal}(L_1∕F_1)\hfill & \hfill \to \hfill & \hfill \mathrm{Gal}(L_2∕F_2)\hfill \\ \hfill \sigma \hfill & \hfill \mapsto \hfill & \hfill \phi \circ \sigma \circ {\phi }^{-1}\hfill \end{array}$$

If $\sigma ,\tau \in \mathrm{Gal}(L_1∕F_1)$,

$$\chi (\sigma )\chi (\tau )=\phi \circ \sigma \circ {\phi }^{-1}\circ \phi \circ \tau \circ {\phi }^{-1}=\phi \circ \sigma \circ \tau \circ {\phi }^{-1}=\chi (\sigma \circ \tau ),$$

so $\chi$ is a group homomorphism.

Moreover, if $\chi (\sigma )=\mathrm{id}$, then $\phi \circ \sigma \circ {\phi }^{-1}=\mathrm{id}$, then $\sigma ={\phi }^{-1}\circ \phi =\mathrm{id}$ : $\ker <!--\; FUNCTION\; APPLICATION\; -->(\chi )=\{\mathrm{id}\}$, so $\chi$ is injective.

If $\tau \in \mathrm{Gal}(L_2∕F_2)$, let $\sigma ={\phi }^{-1}\circ \tau \circ \phi$, then $\sigma \in \mathrm{Gal}(L_1∕F_1)$ with the same arguments, and $\chi (\sigma )=\tau$, thus $\chi$ is surjective.

Conclusion: $\chi :\mathrm{Gal}(L_1∕F_1)\to \mathrm{Gal}(L_2∕F_2)$ is a group isomorphism.

(b)

Suppose as in Proposition 6.1.11 that the restriction of $\phi$ to $F$ is identity, and let $F_1=F$. Then $F_2=\phi (F_1)=F_1=F$, and part (a) shows that

$\chi :\mathrm{Gal}(L_1∕F)\to \mathrm{Gal}(L_2∕F),\sigma \mapsto \phi \circ \sigma \circ {\phi }^{-1}$ is a group isomorphism: this is Proposition 6.1.11.