Calculate the sum of the elements of the U4 set

[Taviii]

I saw this problem in a book: calculate the sum of the elements of the U4 set.

The answer is 0, the elements of the sets being: 1, i, -1, -i.

My questions is: what's the U set?

 

[maverick280857]

In this terminology $U_{n}$ refers to the set of the n-th roots of unity, i.e. all roots of the equation

$$x^{n} - 1 = 0$$

So

$U_{3}$ = {1, $\omega$,$\omega^2$}

 

[Taviii]

Thank you!

Your explanation was very helpful.

 

[HallsofIvy]

If you multiply out (x-a₁)(x-a₂)...(x-a_n) the coefficient of x is easily seen to be -(a₁+ a₂+ ...+ a_n) so for any polynomial equation in which there is no "x" term, the sum of the roots must be 0. In particular, the sum of the roots of xⁿ= 1 must be 0 for all n and so the sum of the elements of U_n must be 0 for all n.