## 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

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

and .

How do you calculate the value of ?

many thanks

sorry again for the inconvenience

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Reply:

Importantly, the correct value is not . This is the complex number on the unit circle at an angle of with the positive real axis. , on the other hand, is a real number. , and are the three solutions to the equation

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

so are and .

As for the value, it depends on the problem. Sometimes, is a perfectly legitimate final expression for a number. On the other hand, you can write it in terms of radicals as

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