**:**

*Complex Roots*If z = r cis(q) is a complex number different from 0, and n is a positive integer, then there are exactly n different complex numbers: w

_{1}, w

_{2}, w

_{3}……w

_{n}each of which is

*the nth ROOT of z.*If we let w = r cis(q) be an nth root of z = rcis(q) so w

^{n}= z

It can eventually be shown ( I leave this to readers) that:

(r cis(q))

^{1/n}= (r)

^{1/n}cis(q/n + k (2 p)/n) , with k = 0,

__+__1,

__+__2…..

*Problem: take the 3rd roots of 1.*

Here: r = 1

The angle is easily determined using and applying the geometry from Galois extensions, for which the number of roots n, divides a unit circle into n equal segments – starting at r = 1 (q = 0 or 2p) and with the angles given at the boundaries of the n segments

With reference to a simple diagram, the reader should be able to sketch the results for n = 3, .i.e. for the 3

^{rd}roots of 1. This yields:

2p/3 = 120 deg and 4p/ 3 = 240 deg

For the first root of unity (k =0) so:

w

_{0}= [1] (cos(0)) = 1

For the 2nd root of unity: (k =

__+__1)

w

_{1}= [1](cos(2p/3) +isin(2p/3)) = - ½__+__i(Ö3/2)
For the 3rd root of unity: (k =

w__+__2)_{2}= [1] (cos(4p/ 3) + isin(4p/ 3) = - ½

__+__i(Ö3/2)

Note that the +/- signs for the roots w

_{1,2}yield two redundant roots. Eliminating them (e.g. removing the secondary sign root duplicates in each case) we have:

w

_{0}= 1

w

_{1}= - ½ + i((Ö3/2)

w

_{2}= - ½ - i((Ö3/2)

*Example Problem*:

Find the
fourth roots of unity and provide a sketch diagram to locate them.

*Solution*: The diagram is

We have, n =
4 with:

w

_{n}= cos (2 p)k/ n + isin(2 pk)/n
For k = 0,
1, 2 and 3 then,

The first root:: w

The first root:: w

_{0}= cos(0) + isin(0) = 1The second root: w

_{1}= cos (p/2) + isin(p/2) = 1i = i

The third root: w

_{2}= cos (p) + isin(p) = -1

And the fourth root:

w

_{3}= cos(3 p/2) + isin(3p/2) = -1i = -i

All of which can be checked using the accompanying diagram which is part of the problem solution.

Note the roots correspond to
successive increases of the angle by p/2 = 90
degrees:

_{}

^{}

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