Exercise 10.1.5

Proof.

(a)

As $\mathrm{Re}({\zeta }_3)=-1∕2$, ${\zeta }_3$ is on the perpendicular bisector of $(-1,0)$.

We gives the details of the construction: $C(0,1)$ and $C(-1,1)$ are constructible by $C2$, so the two intersection points are constructible by $P3$, and these two points are ${\zeta }_3,{\bar{\zeta }}_3$, so ${\zeta }_3$ is constructible.

(b)

As the minimal polynomial of ${\zeta }_3$ over $\mathbb{Q}$ is $x^2+x+1$, the extension $\mathbb{Q}\subset \mathbb{Q}({\zeta }_3)$ is quadratic, so ${\zeta }_3$ is constructible by Theorem 10.1.6.