Norm and Trace of an algebraic number

hbghlyj

Recall that the norm of an algebraic number $\alpha$ is just the product of all the roots of the minimal polynomial, while the trace is the sum of all the roots. Denote the norm of $\alpha$ by $N(\alpha)$ and the trace by $T(\alpha)$. For an algebraic integer, the norm is just the constant term of the minimal polynomial multiplied by $(-1)^{d}$, where $d$ is the degree of the minimal polynomial. The trace is $-1$ times the coefficient of the term of degree $d-1$.
Let $\zeta_{n}:=\exp (2 \pi i / n)$. Show that if $p$ is an odd prime, then $N\left(\zeta_{p^{k}}\right)=1$ and $N\left(\zeta_{p}-1\right)=p$

Czhang271828

The minimal polynomial ($\mathbb Q$) of $\zeta_n$ is $\Phi_n$.

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