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Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ \sigma ( w_1) = w_i$.


I know the converse is true, that is given any $ \sigma\in Aut \mathbb Q(w_1)$, $ \sigma (w_1) $ always maps to some $ w_i$. But for this direction I have trouble on construct the automorphism. Any help is appreciated.

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  • $\begingroup$ Well, for any automorphism $\sigma$ of a field containing $\Bbb Q$, what is $\sigma(r)$ when $r$ is a rational number? And if an automorphism $\sigma$ of $\Bbb Q(w_1)$ satisfies $\sigma(w_1)=w_i$, what are $\sigma(w_2),...,\sigma(\w_{\phi(n)})$? (What is the relationship between $w_1$ and the other $w_j$?) $\endgroup$ Apr 17, 2014 at 7:09
  • $\begingroup$ Well, it's clear that $\sigma (r) = r$ when $ r$ is rational. And I think I can get other $w_j$ by rising $w_1$ to some power, but I don't know how... $\endgroup$
    – user112564
    Apr 17, 2014 at 7:14
  • $\begingroup$ @GregMartin Sorry, forgot to @ you. $\endgroup$
    – user112564
    Apr 17, 2014 at 7:23
  • $\begingroup$ Indeed, the primitive $n$th roots of unity in $\Bbb C$ are $e^{2\pi i k/n}$ for all $1\le k\le n$ such that $\gcd(k,n)=1$. Finding a power $p$ to raise $e^{2\pi i j/n}$ to to get $e^{2\pi i k/n}$ is equivalent to solving $pj\equiv k\pmod n$ for $p$, which is possible since $\gcd(j,n)=1$. $\endgroup$ Apr 17, 2014 at 19:35
  • $\begingroup$ @GregMartin Thanks. And this is actually as far as I reach. I have trouble solving that equation. Can you give any hint? $\endgroup$
    – user112564
    Apr 17, 2014 at 19:43

1 Answer 1

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Your $w_i$ are the roots of $\Phi_n$, the $n$-th cyclotomic polynomial. So your statement is then equivalent to the irreducibility of $\Phi_n$ over $\mathbb Q$ (if $w$ is any root of $\Phi_n$, ${\mathbb Q}[w]$ is isomorphic to the quotient ring $\frac{{\mathbb Q}[X]}{\Phi_n(X)}$).

Do you already know that $\Phi_n$ is irreducible ? If not, you might take a look here for example.

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