- #1
Artusartos
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Homework Statement
I was trying to figure out whether or not ##\zeta_5 + \zeta_5^2## and ##\zeta_5^2 + \zeta_5^3## were complex (where ##\zeta_5## is the fifth primitive root of unity).
Homework Equations
The Attempt at a Solution
##\zeta_5 + \zeta_5^2 = \cos(2\pi/5) + i\sin(2\pi/5) + (\cos(2\pi/5) + i\sin(2\pi/5))^2 = \cos(2\pi/5) + i\sin(2\pi/5) + \cos(4\pi/5) + i\sin(4\pi/5)##.
Since ##i\sin(2\pi/5)## and ##i\sin(4\pi/5)## do not cancel out each other, ##\zeta_5 + \zeta_5^2## must be complex, right?
##\zeta_5^2 + \zeta_5^3 = (\cos(2\pi/5) + i\sin(2\pi/5))^2 + (\cos(2\pi/5) + i\sin(2\pi/5))^3 = \cos(4\pi/5) + i\sin(4\pi/5) + \cos(6\pi/5) + i\sin(6\pi/5)##
But again, the complex numbers don't cancel out each other, right?