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Finding roots of unity using Euler and De Moivreś

Nasser M. Abbasi

June 14,2006 page compiled on July 2, 2015 at 1:28am

To find the roots of

f (x) = xn − 1

Solving for x  from

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Now  1
1n   is evaluated. Since

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Substituting (2) in the RHS of (1) gives

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Using De Moivre's formula

                       (         )        (         )
                1n            2πk-               2πk-
(cos α + isin α ) =  cos  α +   n   +  isin   α +  n         k =  0,1,⋅⋅⋅n − 1

Therefore (3) is rewritten as

          (         )       (         )
                2πk-               2πk-
x = − cos  π +   n    − isin  π +   n         k = 0,1,⋅⋅⋅n − 1

The above gives the roots of          n
f(x) = x  − 1  . The following examples illustrate the use of the above.

1.
Solve         2
f(x) = x  − 1  . Here n =  2  , therefore k = 0,1  . For k =  0
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And for k = 1

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Hence the two roots are {1,− 1}

2.
Solve         3
f(x) = x  − 1  . Here n =  3  , hence for k = 0
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And for k = 1

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And for k = 2

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Therefore the roots are {1,   √ --        √ --
12i 3 − 12,− 12i  3 − 12}