# Help with FST

1. Apr 11, 2007

### bosox097

How do you do these two problems?

1. Find the sum of the 6th roots of unity.
2. Find the product of the 6th roots of unity.

2. Apr 11, 2007

### Data

The real 2nth roots of unity, for any natural n, are -1 and 1. If a complex number is an mth root of unity (for any m) then its complex conjugate is as well. If $z \in \mathbb{C}$ then $z\overline{z} = |z|^2$. Thus the product of the 2nth roots of unity (for any n) is -1.

Furthermore, -z is a 2nth root of unity whenever z is. Thus the sum of the 2nth roots of unity (for any n) is 0.

I hope this wasn't a homework problem!

Last edited: Apr 11, 2007
3. Apr 11, 2007

### robert Ihnot

You should look at symetric functions. Take the cubic: (x-a)(x-b)(x-c)=0. Then if this is multiplied out, we get

$$X^3-X^2(a+b+c)+X(ab+ac+bc)-(abc) = 0.$$

4. Apr 11, 2007

### mathwonk

look at the vertices of a regular hexgon and think of vector addition, and then use group theory.

5. Apr 12, 2007

### HallsofIvy

(x-a)(x-b)= x2- (a+ b)x+ ab
(x-a)(x-b)(x-c)= x3- (a+ b+ c)x2+ (ab+bc+ ac)x- abc
(x-a)(x-b)(x-c)(x-d)= x4- (a+ b+ c+ d)x3+ (ab+ac+ ad+ bc+ bd+ cd)x2- (abc+ acd+ bcd)x+ abcd

Do you see the pattern?

Even more simply: the nth roots of unity are equally spaced around the unit circle in the complex plane. What does symmetry tell you about their sum?