Find roots of polynomial

Question

Let p be an odd prime number and $ζ =ζ_p = cos(2π/p)+isin(2π/p)$

How do you find all of the roots of the polynomial $f(x) = x^{p-1} +x^{p-2}+…+x+1$
How do you show that $p = (1-ζ)(1-ζ^2)…(1-ζ^{p-1}) $
How do you show that $(1-ζ^2)/(1-ζ)$ is a unit in $Z[ ζ ] = (a_0 + a_1ζ+…+a_{p-2}ζ^{p-2} |a_i ∈ Z$)

Knowing $ζ = ζ^{((p+1)/2)·2}$

How do you solve this?

Answer 1

Answer/Hint

Notice that

$$f(x)(x-1)=x^{p}-1$$ so since $f$ is a polynomial of degre $p-1$ and $1$ isn't a root then the other $p$-roots of the unity

$$\zeta_k=\exp\left(\frac{2i k\pi}p\right)=\zeta_p^k,\quad k=1,\ldots,p-1$$

are the roots of $f$.

Moreover, factor $f$ and what's $f(1)$?

Answer 2

$\dfrac{1-\zeta^2}{1-\zeta}$ is a unit because $1-\zeta^2$ and $1-\zeta$ both have norm equal to $p$, hence the norm of the quotient is equal to $1$.

Indeed $N(1-\zeta)=\prod\limits_{k=1}^{p-1}(1-\zeta^k)$ and $N(1-\zeta^2)=\prod\limits_{k=1}^{p-1}(1-\zeta^{2k})$. Since $p$ is prime, any power of $\zeta$ is a generator of the group of $p$-th roots of unity, so that the family $(1-\zeta^{2k})_{1\le k\le p-1}$ is just a permutation of $(1-\zeta^k)_{1\le k\le p-1}$.