complex roots of unity (1 Viewer)

[hatty]

hey, i need help with this please

if w (omega) is a complex root of z^5-1=0
show that w^2 , w^3 and w^4 are the other complex roots.

is this just doing the cis[2k(pi)/n] thing?



and then it says
prove that 1+w+w^2+w^3+w^4 = 0

is this matching the w's to the z's
then doing sometihng like

z^5-1 = 0

(z - 1)(z^4+z^3+z^2+z^1+z) = 0
?


thanks

 

[grimreaper]

yep you were right about everything you said however since w is not equal to 1, 1+w+w^2+w^3+w^4 = 0

 

[hatty]

thanks mate

 

[CM_Tutor]

You can show that w^2, w^3 and w^4 are the other roots without using the "cis[2k(pi)/n] thing", as you put it.

We know w is a root, and so w^5 = 1.

Now, test z = w^2: LHS = z^5 - 1 = (w^2)^5 - 1 = w^10 - 1 = (w^5)^2 -1 = 1^2 - 1 = 0 = RHS.

Similarly for w^3, w^4.

Now you only need to note that these are different (which they obviously are. For example, w = w^2 only if w = 1 or w = 0, neither of which is the case here, and similarly for w^3 and w^4).

Thus, the 5 solutions of z^5 -1 = 0 are 1, w, w^2, w^3 and w^4, where w is any root that isn't 1.