Cyclotomic Polynomial Questions

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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?
 

Answers and Replies

  • #2
andrewkirk
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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.
 
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  • #3
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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.
 
  • #4
andrewkirk
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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##?
 
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