Exercise 10.2.4

Proof.

$\text{∙}$

Suppose that $m\mid n,m>0$. Since $n∕m$ is an integer, $${({\zeta }_n)}^{\frac{n}{m}}={\left(e^{\frac{2\mathit{\pi i}}{n}}\right)}^{\frac{n}{m}}=e^{\frac{n}{m}\frac{2\mathit{\pi i}}{n}}=e^{\frac{2\mathit{\pi i}}{m}}={\zeta }_m.$$

$\text{∙}$

If ${\zeta }_n$ is constructible, and $m\mid n,m>0$, by part (a) $${\zeta }_m={({\zeta }_n)}^{\frac{n}{m}}$$

is a power of ${\zeta }_n$.

As the set of constructible points is a subfield of $ℂ$, ${\zeta }_m$ is constructible.