Exercise 15.5.3

Proof. Let $n\ge 1$ be an integer. Suppose first that $n$ is odd.

By the proof of Theorem 15.5.1, the splitting field of $A_n=u{P}_n(u^4)$ is $L=\mathbb{Q}\left(i,\phi \left(\frac{\varpi }{n}\right)\right)$ and $\mathrm{Gal}(L∕\mathbb{Q}(i))$ is Abelian by this Theorem 15.5.1, therefore $\mathrm{Gal}(L∕\mathbb{Q}(i))$ is solvable.

Thus the Galois group of $A_n(u)=u{P}_n(u^4)\in \mathbb{Q}(i)[u]$ is solvable by radicals over $\mathbb{Q}(i)$. This implies that all the roots of $A_n(u)$ are solvable by radicals over $\mathbb{Q}(i)$, and since $\mathbb{Q}\subset \mathbb{Q}(i)$ is radical, all the roots of $A_n(u)$ are solvable by radicals over $\mathbb{Q}$.

Section 15.2 shows that the polar distances of the $n$-division points are

and, since $n$ is odd,

where $P_n$ and $Q_n$ have no common root, thus $r_m$ is a root of $A_n(u)=u{P}_n(u^4)$, and $r_m$ is expressible by radicals for all $m\in \mathbb{Z}$.

If $x,y$ are the coordinates of the $m$-th division point, then the equation of the lemniscate ${(x^2+y^2)}^2=x^2-y^2$ gives

thus

so that $x^2,y^2\in \mathbb{Q}(r_m)$. This proves that $x,y$ are expressible by radicals.

If $n$ is even, then $n=2^s{n}^{\prime }$, where $n^{\prime }$ is odd. The first part shows that

are expressible by radicals.

If we replace the word ”constructible” in the proof of Proposition 15.2.3 by ”expressible by radicals”, we see that $r_0=\phi \left(\frac{x_0}{2}\right)$ is expressible by radicals if $a=\phi (x_0)$ is expressible by radicals: the equations (15.15) and (15.16) show that there is some $t\in ℂ$ such that

Since $t^2$ in a quadratic extension of $\mathbb{Q}(a)$ and $r_0^2$ is in a quadratic extension of $\mathbb{Q}(t)$, the chain $\mathbb{Q}\subset \mathbb{Q}(i)\subset \mathbb{Q}(i,a)\subset \mathbb{Q}(i,a,t)\subset \mathbb{Q}(i,a,t,r_0)$ shows that $r_0$ is expressible by radicals over $\mathbb{Q}$.

Repeating $s$ times this argument, we see that the polar distances of the $n$-division points

are expressible by radicals for all $m\in \mathbb{Z}$, and as above the coordinates of the $m$-division points are expressible by radicals. □