1. ## Two algebra questions

I have problem with these two exercise:

1. Assume, that $\displaystyle \epsilon \in C$ is a nth root of unity. Show: if $\displaystyle \epsilon \neq 1$, then $\displaystyle \epsilon$ is the root of the syclotomic polynomial $\displaystyle \Phi_n = 1 + X + X^2 + \dots + X^{n-1} \in \mathbb{C}[X]$.
($\displaystyle \epsilon = e^{2\pi i / n}$)

2. Show: if $\displaystyle k \in \mathbb{C}$, then $\displaystyle X^n - k^n = (X-k)(X-\epsilon k) \cdots (X-\epsilon^{n-1}k)$, where $\displaystyle \epsilon = e^{2\pi i / n}$.

I have no idea how to begin with these, so any help would be nice.

2. Hints:

1. $\displaystyle X^n-1=(X-1)\Phi_n(X)$ .

2. Clearly, $\displaystyle k$ is a root of $\displaystyle X^n-k^n$ . Prove that $\displaystyle \epsilon^jk$ is also a root for every $\displaystyle j=1,\ldots,n-1$

Fernando Revilla

3. Further to FernandoRevilla's reply, basically an n-th root of unity is a corner of a regular n-gon inscribed in the uni circle.

For example,

This is because it must satisfy the equation $\displaystyle x^n-1=0$. That is where the $\displaystyle \epsilon=e^{2 \pi i/n}$ comes from.