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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?

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    $\begingroup$ I edited your post to make the $\LaTeX$ work. Remember, "\$" and $\{ \cdot \}$ are your two best friends in $\LaTeX$. Cheers! $\endgroup$ – Robert Lewis Dec 2 '18 at 23:22
  • $\begingroup$ Did you show that $\Phi_p(t) = \sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $\zeta_p$ ? $\endgroup$ – reuns Dec 3 '18 at 0:12
  • $\begingroup$ 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. $\endgroup$ – Lubin Dec 3 '18 at 22:05

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