Unit in cyclotomic field Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can anything be said about that?
 A: This is only a partial answer: it reduces the question to another problem.
Taking up Ilya Bogdanov's comment, let $a$ be a root of $2x^2 + x + 2$.
Then (as he says) we are interested in the set of numbers $n$ such that
$2^{\varphi(n)/2} |\Phi_n(a)| = 1$. We note that
$t_n := (2^{n/2} |a^n - 1|)^2 = 2^{n+1} - \operatorname{Tr}(2a)^n$
satisfies the recurrence $$t_{n+3} = t_{n+2} - 2 t_{n+1} + 8 t_n$$
with initial values $t_0 = 0$, $t_1 = 5$, $t_2 = 15$.
Now the question is, for which $n$ is
$$N(n) := \prod_{d\mid n} t_d^{\mu(n/d)} = 1 \quad ?$$
If one could show that for large enough $n$, $t_n$ always has
a primitive prime divisor $p$ (meaning a prime $p$ such that $p$ divides 
$t_n$, but $p$ does not divide $t_m$ for any $1 \le m < n$), then it
would follow that $N(n) > 1$ for all large $n$ (since $p$ divides $N(n)$).
However, I am not aware of any general results on primitive prime divisors
of ternary recurrences. There are quite general statements for
binary recurrences, for example

Bilu, Yu.; Hanrot, G.; Voutier, P. M.:
  Existence of primitive divisors of Lucas and Lehmer numbers.
  J. Reine Angew. Math. 539 (2001), 75-122.

where it is shown that in this case there is always a primitive prime
divisor when $n > 30$.
Proving a similar statement even for the specific sequence $(t_n)$ might
be hard.
