# Galois theory: Gauss-Wantzel theorem, proof explanation

I am using Ian Stewart Galois theory book

and it says

1. that for $$A =$$ primitive $$p^2$$ root of unity $$A$$ has min poly of $$m(t)= 1+ t^p +.....+t^{p(p-1)}$$ and so $$p(p-1)$$ is a power of two. why is this so?

2. if $$p^k$$ -sided regular polygon is constructive, then so must $$p^2$$ - sided regular polygon, for $$k \ge 2$$. why is this so?

• I edited your post to make the $\LaTeX$ work. Remember, "\$" and$\{ \cdot \}$are your two best friends in$\LaTeX$. Cheers! – Robert Lewis Dec 2 '18 at 23:22 • Did you show that$\Phi_p(t) = \sum_{n=0}^{p-1} t^n$is the minimal polynomial of$\zeta_p$? – reuns Dec 3 '18 at 0:12 • 1. It can’t be that$p(p-1)$is a power of$2$: Fundamental Theorem of Arithmetic says that in that case, both$p$and$p-1$would be powers of$2$. Unless$p-1$=1, of course. For 2, note that not even the$9\$-sided regular polygon is constructible. – Lubin Dec 3 '18 at 22:05