4
$\begingroup$

Consider the polynomial $x^n - 1$, then a splitting field for $x^n- 1$ over $\mathbb{Q}$ is $Q(\omega)$ where $\omega$ is the nth primitive root of unity. If we look at the values we can send $\omega$ to that keeps the criteria that whatever we send it to is also an nth root of unity, then it seems that we can send $$\omega \rightarrow \omega, \omega^2, \omega^3, ..., \omega^{n-1}$$ Because $(\omega^k)^n = 1$ where $k < n$. To me, it seems that this is perfectly valid and these are the only automorphisms for $\mathbb{Q}(\omega)$ that fix $\mathbb{Q}$, which implies that $|\text{Gal}(\mathbb{Q}(\omega)/\mathbb{Q})| = n - 1$, but I'm almost sure I'm wrong, because I know that $$\text{Gal}(\mathbb{Q}(\omega)/\mathbb{Q}) \approx U(n)$$ And it is not necessary for $|U(n)| = n - 1$, so where am I going wrong?

$\endgroup$
7
  • 1
    $\begingroup$ When $n=2$, $\omega=-1$ and $1$ is also a second root of unity; is there an automorphism mapping $-1$ to $1$? $\endgroup$ Commented Dec 9, 2017 at 7:53
  • 1
    $\begingroup$ You've only listed $n-1$ maps though? $\omega$ cannot go to $1$, since the latter is rational. A very useful case to study is $n=4$, where $\omega=i$. $\endgroup$
    – Steve D
    Commented Dec 9, 2017 at 7:54
  • $\begingroup$ My apologies, I made a counting mistake, I have edited the question $\endgroup$
    – q.Then
    Commented Dec 9, 2017 at 7:56
  • 1
    $\begingroup$ When $n=4$, $\omega=i$, and $-1$ is also a fourth root of unity; is there an automorphism mapping $i$ to $-1$? $\endgroup$ Commented Dec 9, 2017 at 8:04
  • $\begingroup$ @LordSharktheUnknown, I've thought about it for a while, and don't understand why can't there be an automorphism mapping $i$ to $-1$? $\endgroup$
    – q.Then
    Commented Dec 9, 2017 at 8:10

1 Answer 1

4
$\begingroup$

You list the possible nontrivial group homomorphisms of $μ_n = \{ζ ∈ ℚ(ω); ζ^n = 1\}$. Of course, every field automorphism on $ℚ(ω)$ restricts to such a homomorphism, but that restriction is then an automorphism as well. So you cannot send $ω$ to each of $ω, …, ω^{n-1}$, but only to those of the same order.

$\endgroup$
2
  • 1
    $\begingroup$ it sounds like $\phi$ needs to send $\omega$ to $\omega^k$, and $(k, n) = 1$? $\endgroup$
    – q.Then
    Commented Dec 9, 2017 at 8:13
  • 1
    $\begingroup$ @q.Then Yes, because $μ_n \cong ℤ/nℤ$ via $ω^k \leftrightarrow k$, and so the elements of order $n$ in $μ_n$ correspond to those $k=1, …, n$ with $(k,n) = 1$. $\endgroup$
    – k.stm
    Commented Dec 9, 2017 at 8:41

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .