root of unity

Dear Dr Kim

I know you must be recieving many emails of this sort at this moment in time, so i’ll try and keep this as brief as possible: just a quick (easy) question,

When trying to calculate the roots of a polynomial say

t^3 -5

I know it can be split into its linear factors where a= cube root of 5

(t-a)(t-aw)(t-aw^2)

and w=exp(2\pi/3).

How do you calculate the value of w?

many thanks
sorry again for the inconvenience

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Reply:

Importantly, the correct value is w=exp(2\pi i/3) not exp(2\pi/3). This is the complex number on the unit circle at an angle of 2\pi/3 with the positive real axis. exp(2\pi/3), on the other hand, is a real number. 1,w, and w^2 are the three solutions to the equation

t^3=1.

For this reason, whenever \alpha is a solution to an equation of the form

t^3-z=0,

so are \alpha w and \alpha w^2.

As for the value, it depends on the problem. Sometimes, exp(2\pi i/3) is a perfectly legitimate final expression for a number. On the other hand, you can write it in terms of radicals as

exp(2\pi i/3)=cos(2\pi/3)+isin(2\pi/3)=-1/2+i\sqrt{3}/2.

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