TITLE: An algebraic number is not a root of unity?
QUESTION [18 upvotes]: This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that boils it down to the following question:

Let $\xi$ be a primitive root of unity of degree $2n>12$ (just in case). Then the roots $\lambda$ of $\chi(\lambda):=\lambda^2+(\xi^2-\xi+1)\lambda+\xi^2$ are not roots of unity.

I was particularly interested in the cases $n=7$ or $9$. In both cases, $\deg\Phi_{2n}=6$ (here $\Phi$ is the cyclotomic polynomial) and, hence, $\mathbb{Q}[\lambda]$ has degree at most $12$. There are finitely many (although quite a few) cyclotomic polynomials $\Phi_k$ with $\deg\Phi_k\le12$. For each such polynomial, one computes the $\lambda$-resultant $R(\xi)$ of $\Phi_k$ and $\chi$ and checks that $R(\xi)\ne0\bmod\Phi_{2n}(\xi)$.
Clearly, this approach should work for any given $n$, but I have no idea how this can be done "in general". So, here is the question:

Is there a smarter way to prove that an algebraic number is not a root of unity?

REPLY [11 votes]: Just noticed this question.
I think the following is an even more elementary/self-contained proof.
First, let $\eta = \xi u$ (so $u$ is a root of unity iff $\eta$ is),
and divide by $\xi^2$ to get
$$
u^2 + (\xi-1+\xi^{-1}) u + 1 = 0,
$$
whose two roots are inverses of each other, so either both of them are
roots of unity or neither is.  Now since all primitive $(2n)$-th roots of
unity are conjugate, $u$ is a root of unity for one of them iff
it is a root of unity for all of them.  But then
$$
\xi - 1 + \xi^{-1} = -(u+u^{-1}) \geq -2,
$$
which holds iff $\xi = e^{2\pi i r}$ with $|r| \leq 1/3$.
But the primitive $(2n)$-th roots of unity are precisely
$e^{2 \pi i r}$ for $r=m/2n$ with $m \in [-n,n]$ coprime to $n$.
So it comes down to checking that there is such $r > 2n/3$.
But $r$ can be as large as $n-1$ or $n-2$ according as $n$ is even or odd,
and that exceeds $2n/3$ once $n>7$, QED.
Conversely, if $n$ is one of the few small numbers such that $2n$
is not coprime with any $r \in (n/3, 2n/3)$ then all conjugates of $u$
are algebraic integers of absolute value 1, and thus roots of unity
by a theorem of Kronecker; you presumably observed this for $n=5$.