Exercise 6.2.3

Proof. Write $\zeta ={\zeta }_5$.

(a)

as $L=\mathbb{Q}(\zeta ,\sqrt[5]{2})$, Proposition 6.1.4(b) shows that $\sigma \in \mathrm{Gal}(L∕\mathbb{Q})$ is uniquely determined by $\sigma (\zeta ),\sigma (\sqrt[3]{2})$.

Moreover by Proposition 6.4.1(a), $\sigma (\zeta )$ is a root of $f=x^4+x^3+x^2+x+1$, whose roots are ${\zeta }^i,1\le i\le 4$, and $\sigma (\sqrt[5]{2})$ is a root of $g=x^5-2$ , whose roots are ${\zeta }^j\sqrt[5]{2},0\le j\le 4$.

Then Exercice 6.1.1 shows that

$$|\mathrm{Gal}(L∕\mathbb{Q})|\le \mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(f)\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(g)=20.$$

Moreover, by Exercise 5.1.8, $[L:\mathbb{Q}]=4\times 5=20$.

(b)

$L$ is the splitting field of the separable irreducible polynomial $g=x^5-2\in \mathbb{Q}[x]$ over $\mathbb{Q}$. Indeed, $g$ is irreducible over $\mathbb{Q}$ by Schönemann-Eisenstein Criterion with $p=2$, and separable since its roots in $ℂ$ are $\sqrt[5]{2},\zeta \sqrt[5]{2},{\zeta }^2\sqrt[5]{2},{\zeta }^3\sqrt[5]{2},{\zeta }^4\sqrt[5]{2}$ which are distinct.

By theorem 6.2.1, $|\mathrm{Gal}(L∕\mathbb{Q})|=[L:\mathbb{Q}]$, therefore

$$|\mathrm{Gal}(L∕\mathbb{Q})|=[L:\mathbb{Q}]=20.$$