Find the product $(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$ Let $1,$ $a_i$ for $1 \leq i \leq 6$ be the different roots of $x^7-1$.
Then find the product:
$(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$
I don't know how to proceed.

We have $$(x-1)(x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5)(x-a_6)=(x-1)(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}).$$ Hence we must have $$(x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5)(x-a_6)=1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}.$$ This is because I am dividing the same monomial $x-1$ from both sides. Hence we have $$\prod_{i=1}^{6}(1-a_i)=7.$$
