Exercise 7.2.10

Proof. Let $L=\mathbb{Q}(\omega ,\sqrt[3]{2})$.

$\tau$ is the unique $\mathbb{Q}$-automorphism of $G=\mathrm{Gal}(L∕\mathbb{Q})$ such as

If $z\in ℂ$ is element of $L$, then $z=p(\omega ,\sqrt[3]{2})$, where $p(x,y)\in \mathbb{Q}[x,y]$, thus $\bar{z}=p(\bar{\omega },\sqrt[3]{2})=p(-1-\omega ,\sqrt[3]{2})\in L$. Let $\lambda :L\to L,z\mapsto \bar{z}$ the restriction (and corestriction) of the conjugation in $ℂ$. Then $\lambda$ is an involutive ring homomorphism, thus an automorphism of the field $L$, which is the identity on $\mathbb{Q}$: $\lambda \in \mathrm{Gal}(L∕\mathbb{Q})$. As

and as a $\mathbb{Q}$-automorphism of $L=\mathbb{Q}(\omega ,\sqrt[3]{2})$ is uniquely determined by the images of $\omega ,\sqrt[3]{2}$, $\tau =\lambda$, so $\tau$ is the complex conjugation restricted to $\mathbb{Q}(\omega ,\sqrt[3]{2})$. □