1. ## cyclotomic polynomials

We know that $\displaystyle \Phi_{p}(x) = \frac{x^p-1}{x-1} = x^{p-1}+x^{p-2}+ \dots + x+1$ is irreducible over $\displaystyle \mathbb{Q}$ for every prime $\displaystyle p$. Suppose $\displaystyle \gamma$ is a zero of $\displaystyle \Phi_{p}(x)$ and consider the field $\displaystyle \mathbb{Q}(\gamma)$.

(a) Show that $\displaystyle \gamma, \gamma^2, \dots, \gamma^{p-1}$ are distinct zeros of $\displaystyle \Phi_{p}(x)$ and show that they are all the zeros of $\displaystyle \Phi_{p}(x)$.

(b) Show that $\displaystyle G(Q(\gamma)/ \mathbb{Q})$ is abelian of order $\displaystyle p-1$.

(c) Show that the fixed field of $\displaystyle G(\mathbb{Q}(\gamma)/ \mathbb{Q})$ is $\displaystyle \mathbb{Q}$.

For (a), these are the nth roots of unity? For (b), consider two automorphisms of $\displaystyle \mathbb{Q}(\gamma)$ and show that they commute? For (c), isn't this by definition?

2. Originally Posted by Sampras
We know that $\displaystyle \Phi_{p}(x) = \frac{x^p-1}{x-1} = x^{p-1}+x^{p-2}+ \dots + x+1$ is irreducible over $\displaystyle \mathbb{Q}$ for every prime $\displaystyle p$. Suppose $\displaystyle \gamma$ is a zero of $\displaystyle \Phi_{p}(x)$ and consider the field $\displaystyle \mathbb{Q}(\gamma)$.

(a) Show that $\displaystyle \gamma, \gamma^2, \dots, \gamma^{p-1}$ are distinct zeros of $\displaystyle \Phi_{p}(x)$ and show that they are all the zeros of $\displaystyle \Phi_{p}(x)$.
First, the zero of $\displaystyle \Phi{p}(x)$ are just the zeros of $\displaystyle x^p- 1$, excluding 1. If if $\displaystyle \gamma$ is such a 0, then $\displaystyle \gamma^p= 1$ so that $\displaystyle (\gamma^n)^p= (\gamma^p)^n= 1$. The only 'hard' part is showin that $\displaystyle \gamma^n$ is not 1 for n= 1, 2, ..., p-1. It is important that p is prime. here.
Suppose $\displaystyle \gamma^i= \gamma^j$ for i< j< p. Then $\displaystyle \gamma^{j- i}= 1$. Show that that is impossible.

[tex](b) Show that $\displaystyle G(Q(\gamma)/ \mathbb{Q})$ is abelian of order $\displaystyle p-1$.

(c) Show that the fixed field of $\displaystyle G(\mathbb{Q}(\gamma)/ \mathbb{Q})$ is $\displaystyle \mathbb{Q}$.

For (a), these are the nth roots of unity? For (b), consider two automorphisms of $\displaystyle \mathbb{Q}(\gamma)$ and show that they commute? For (c), isn't this by definition?[/QUOTE]