Constructing intersections in a regular polygon

Question

Consider a regular polygon with $n$ vertices. The position of the vertices are given by $e^{2 i \pi k / n}$ for $k \in \{1 \ldots n\}$. The vertices are in $\mathbb{R}[i]$ but they are also in $\mathbb{Q}[x]/\langle \Phi_n(x) \rangle$ where $\Phi_n$ is the nth cyclotomic polynomial.

In this field, the vertices can be represented as $1, x, \ldots, x^{n-1}$.

Consider the lines going through, respectively, $x^a$ and $x^b$ and $x^c$ and $x^d$, and let them intersect at point $u$.

1. Does $u \in \mathbb{Q}[x]/\langle \Phi_n(x) \rangle$? If not, what field extension do we need?
2. How would one go about computing $u$?

Answer 1

Let $(x^a, x^b)$ and $(x^c, x^d)$ be the two segments; if they intersect they do so at:

$$\frac{x^{-a}+x^{-b}-x^{-c}-x^{-d}}{x^{-a-b}-x^{-c-d}}$$