# About a root in the topic of automorphism and fixed field. HELP

1. Jun 1, 2012

### julypraise

About a root in the topic of automorphism and fixed field. HELP!!

1. The problem statement, all variables and given/known data
Let $m$ be a positive integer.

Let $\xi = \exp (2pi/m)$. Then $\xi$ is a primitive m-th root of unity. (I.e., $\xi$ is a solution of

$\Phi_{m}(X):=(X^m - 1)/(X-1)$.)

If $\phi \in \mbox{G}(\mathbb{Q}(\xi)/\mathbb{Q})$, i.e., $\phi$ is an automorphism of $\mathbb{Q}(\xi)$ fixing the elements in $\mathbb{Q}$, then $\phi (\xi) = \xi^i$ for some $i$ with $\gcd (i,m) = 1$

2. Relevant equations

$\mathbb{Q}(\xi) = \mathbb{Q}(\xi^i)$ for any $i$ with $\gcd(i,m)=1$ but not with $i$ such that $\gcd(i,m) \neq 1$

3. The attempt at a solution

I know the above relevant equation holds. And also I know that $\phi(\xi) = \xi^i$ for some integer $i$ but I cannot prove $\gcd (i,m) = 1$. I've tried to use Isomorphism Extension Theorem, but not really works.

Last edited: Jun 1, 2012