Let $w$ be a primitive $p$th root of unity and consider the equation $$w^x+w^y+w^z=w^{x+y}+w^{y+z}+w^{z+x},$$ where $x,y$ and $z$ are integers.
Cyclotomic polynomials can be used to settle many cases. For example, for $p$ prime, a solution requires the multiset {$x,y,z$} to be of the form {$0,t,-t$}, modulo $p$.
Is there a neat way to solve the equation for general $p$?
Let $e(x)=\exp(2\pi i x)$ and $U=e({\mathbb Q})$. Putting $a=w^x,b=w^y,c=w^z$, your question can be rephrased as : find all $a,b,c\in U$ such that
$$a+b+c=ab+ac+bc \tag{1}$$.
Now, put $u_1=a,u_2=b,u_3=c,u_4=-ab,u_5=-ac,u_6=-bc$. We have :
Theorem. Let $u_1,u_2,\ldots,u_6 \in U$. Then $u_1+u_2+\ldots +u_6=0$ iff for some permutation $\sigma$ of $[|1..6|]$, and $v=(u_{\sigma(1)},u_{\sigma(2)},\ldots,u_{\sigma(6)})$ one of the following two holds :
(i) $v=(\rho_1,-\rho_1,\rho_2,-\rho_2,\rho_3,-\rho_3)$, with $\rho_1,\rho_2,\rho_3 \in U$
(ii) $v=(\rho_1,\rho_1 e(\frac{1}{3}),\rho_1 e(\frac{2}{3}), \rho_2,\rho_2 e(\frac{1}{3}),\rho_2 e(\frac{2}{3}))$, with $\rho_1,\rho_2 \in U$.
(iii) $v=(\rho e(\frac{1}{5}),\rho e(\frac{2}{5}), \rho e(\frac{3}{5}), \rho e(\frac{4}{5}),-\rho e(\frac{1}{3}),-\rho e(\frac{2}{3}))$, with $\rho \in U$.
Proof : see see Henry B. Mann, "On linear relations between roots of unity", Mathematika 12(1965), pp.107-117.
For shorter formulas, let us put $j=e(\frac{1}{3}),\eta=e(\frac{1}{5})$. (i) yields the following solutions to (1) :
$$ \lbrace a,b,c \rbrace = \lbrace x,-x,-x^{2} \rbrace, \lbrace 1,x,x^{-1} \rbrace, \ \textrm{or} \ \lbrace y_1,y_2,y_1y_2 \rbrace, \ \textrm{with} \ x\in U,y_1=\pm i, y_2=\pm i. \tag{2} $$
(ii) yields the following solution to (1) :
$$ \lbrace a,b,c \rbrace = \lbrace x,jx,j^2x\rbrace \ \textrm{or} \ \lbrace x,-j,-j^2\rbrace, \ \textrm{with} \ x\in U. \tag{3} $$
(iii) yields the following solutions to (1) :
$$ \lbrace a,b,c \rbrace = \lbrace -\eta^x, j^{y}\eta^{2x},j^{y}\eta^{4x}\rbrace, \ \textrm{with} \ 1 \leq x \leq 4, 1\leq y \leq 2. \tag{4} $$