I am reading "Notes in connexion with Fermat's last theorem", by G. B. Mathews. (accessible per example on Google books)
For some integer $k$, he defines $r = e^{\frac{2 \pi i}{k}}$ and: $$ P_k(r) = \Pi_{\alpha, \beta, \gamma} r^\alpha + r^{\beta} + r^{\gamma} $$
Where the products run over $r^\alpha, r^\beta$ and $r^\gamma$, distinct (complex) roots of $x^k - 1 = 0$.
He then claims that $P_k(r)$ is the $k$th power of an integer : $P_k(r)=u_k(r)^k$ (I am not sure why, but I will leave this part to another question).
For $q$ prime number of the form: $q=nk+1$, he then defines $x_\alpha,x_\beta$ and $x_\gamma$ (integers) the roots of $x^k \equiv 1 \mod q$.
And then he states that a necessary condition for:
$$x_\alpha+x_\beta+x_\gamma\equiv0 \mod q$$
Is $u_k(r) \equiv 0 \mod q$. I do not really get this point. Is there a theorem that directly states the following?
$$x_\alpha \equiv r^\alpha \mod q$$
This way we would obtain : $x_\alpha+x_\beta+x_\gamma\ \equiv r^\alpha + r^\beta + r^\gamma \mod q$ and the condition on $u_k$ would follow.
Edit
After a second thought, the theorem as stated above is way too optimistic.
However, the following theorem seems to solve the issue (Edwards, Fermat's last theorem, a genetic introduction to algebraic number theory).
If $h(r)$ is a cyclotomic integer which divides both $x+r^iy$ ($x$ and $y$ assumed coprime, $r^i \neq 1$) and $p$ (a prime integer), then there exists and integer $l$ such that $r \equiv l \mod h(r)$ and such that:
$$f(r)\equiv 0 \mod h(r) \iff f(l)\equiv 0 \mod p$$
Assuming, in our case that $q$ has a factor $h(r)$, then:
$u_k(r) \equiv 0 \mod q \implies u_k(r) \equiv 0 \mod h(r)$
By the above theorem:
$u_k(r) \equiv 0 \mod h(r) \implies u_k(l) \equiv 0 \mod q$ (for some integer $l$).
Now if $u_k(l) \equiv 0 \mod q$ it means that one the elements of the product: $\Pi_{\alpha, \beta, \gamma} l^\alpha + l^{\beta} + l^{\gamma}$ is divisible by $q$.
Similarly, we check that $l^k \equiv 1 \mod q$
And we conclude that $l^\alpha, l^\beta, l^\gamma$ are all $n$th powers.
My questions (after edit) are
Is my reasonning correct ?
How can I be sure of the existence of $h(r)$ ?
The theory that $ x_\alpha \equiv r^{\alpha'} \mod q$ is indeed to optimistic as $r^{\alpha'}$ in not in even a integer in almost every case (1 and -1 are the only integer unit roots).
Then there is a problem when you asume $q$ has a factor $h(r)$ as $q$ is prime, so unless $h(r) \in \{\pm1,\pm q\}$, the following to that is not correct.
I had an idea, but i am unable to finish it at the moment, to use the relation between $k-$unit roots, denoted as $\mathbb{C}_k$ and intergers in module $k$, denoted $\mathcal[Z]_k$, given by the isomorphism:
$\begin{align*}\varphi: \mathcal[Z]_k &\to \mathbb{C}_k\\ 1 &\to r = e^{\frac{2 \pi i}{k}} \end{align*}$
( The group operation in $\mathbb{C}_k$ is the multiplication in $\mathbb{C}$)
And use that to make arguments.
I will be thinking further more this.