Abstract Algebra. Find the number of elements in the cyclic subgroup of the group C* generated by (1+i)/√2?

C* is the group of nonzero complex numbers under multiplication. Suggestion: Convert to polar coordinates

Abstract Algebra. Find the number of elements in the cyclic subgroup of the group C* generated by (1+i)/√2?
Okay, let's follow the suggestion and convert to polar coordinates:





abs((1+i)/sqrt(2))


= sqrt((1+i)(1-i)/2)


= sqrt((1 - (-1)) / 2)


= sqrt(1)


= 1,





and the angle is (obviously) 45 degress (or pi/4), because 1 and i have the same length.





Let's set z = (1+i)(1-i)/sqrt(2),





we just saw that z = exp(2 pi i / 8),





and we see z has order 8, i.e. the generated cyclic subgroup of C* has order 8.





Geometrically, the generated subgroup is an octagon, placed on the unit circle in the complex plane.

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