Computer Science or Algebra?

Algebra Level 4

What values of z z satisfy the equation below?

z + 1 z = z 2 + 1 z 2 = z 4 + 1 z 4 = ⋯ = z 2 m + 1 z 2 m = ⋯ z+\frac{1}{z}=z^2+\frac{1}{z^2}=z^4+\frac{1}{z^4}=\cdots=z^{2^m}+\frac{1}{z^{2^m}}=\cdots

Insufficient information ± ω \pm \omega , ± ω 2 \pm \omega^2 + 1 +1 , + ω +\omega , + ω 2 +\omega^2 ± 1 \pm 1 , ± ω \pm \omega , ± ω 2 \pm \omega^2 − 1 -1 , − ω -\omega , − ω 2 -\omega^2 No such value exists None of these − 1 -1 , + 1 +1

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Zeeshan Ali by Zeeshan Ali , 25, Pakistan

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1 solution

Dong kwan Yoo
May 30, 2018

By condition of problem,

solution of x + 1 x x + \frac{1}{x} = x 2 + 1 x 2 x^2 + \frac{1}{x^2 } : z , z 2 , z 4 , . . . . z ~,~z^2 ~,~z^4 ~,~ ....

x + 1 x x + \frac{1}{x} = x 2 + 1 x 2 x^2 + \frac{1}{x^2 }

x 3 + x = x 4 + 1 x^3 + x = x^4 + 1

( x − 1 ) 2 ( x 2 + x + 1 ) = 0 (x-1)^2 (x^2 + x + 1 ) = 0

therefore x = ω , ω 2 , 1 x = \omega , \omega ^2 , 1

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