Cyclotomic Polynomial Questions

[PsychonautQQ]

Homework Statement

I have a series of questions regarding a cyclotomic polynomial of order p-1 where p is a prime; so there are p total terms in this polynomial because their is a constant term. I will post the questions 1 at a time, and as soon as I work my way through one i'll post the next one in this same thread.

Let's go !

First Question: If b is a root of this cyclotomic polynomial, show that b, b^2, ... b^p-1 are all distinct roots of this polynomial

Homework Equations

The Attempt at a Solution

I'm at a bit of a loss as where to start, Perhaps it is of note that this polynomial is irreducible, and it is the pth cyclotomic polynomial (although this means it is of order p-1 where p is a prime). I suspect that these two factoids are going to play a role in the solution to understanding why any root will generate all the other roots. Anyone have some insight for me?

 

[andrewkirk]

I suggest starting with the formula given at the start of the wiki article. Since $p$ is prime, the formula becomes
$$\Phi_{p}=\prod_{k=1}^{p-1}\Big(x-e^{2\pi i\frac kp}\Big)$$

You can use the primeness of $p$ to show that any root $e^{2\pi i\frac kp}$ can generate all the others.

 

[PsychonautQQ]

Can you give me a hint on how I would show this? I understand that in the multiplicative group Z mod p, every element is a generator, and I'm aware that this must be pretty directly related to that fact, however I'm failing to completely connect the dots. The fact that p is in the denominator of the exponent is throwing me off I think, making it hard for me to see why this must be.

 

[andrewkirk]

You're on the right track. I can't give much of a hint without giving the whole thing away. But if you think about the following two questions, there's a good chance you'll see a promising way forward.

1. If $e^{2\pi i \frac kp}=e^{2\pi i \frac lp}$ what can we say about the relationship between $k$ and $l$?
2. If $b=e^{2\pi i \frac kp}$ and $r$ is an integer between $1$ and $p-1$ inclusive, what is $b^r$, written as a power of $e$?