If $\alpha_1, \alpha_2, \dots, \alpha_{100}$ are all the 100th root of unity, then what is the numerical value of ${\displaystyle\sum\displaystyle\sum}_{1\leq i<j\leq 100} (\alpha_i\alpha_j) ^5$?
added
We have
${\displaystyle\sum\displaystyle\sum}_{1\leq i<j\leq 100} (\alpha_i\alpha_j) ^5={\displaystyle\sum\displaystyle\sum}_{1\leq i<j\leq 100} \alpha_i^5\alpha_j^5$. I don't know what to do now.
Since $\alpha^{100}=1$, we see that our sum equal to $0$
because by the Viete's theorem $$\sum_{i=1}^{100}\alpha_i=\sum_{1\leq i<j\leq100}a_ia_j=...=\sum_{1\leq i_1<i_2<...<i_{10}\leq100}a_{i_1}a_{i_2}...a_{i_{10}}=0$$