Let G be the set of the fifth roots of unity Prove that G is

Let G be the set of the fifth roots of unity. Prove that G is isomorphic to Z5 under addition by doing the following: a. State each step of the proof. b. Justify each of your steps of the proof.

Solution

G is the set of 5th root of unity

If z⁵ =1 = cos 2npi+isin2npi

Then z has values as

1, cos 2pi/5+isin2pi/5, cos 4pi/5+isin 4pi/5, cos 6pi/5+isin 6pi/5, cos 8pi/5+isin 8pi/5

Let Z5 = (0,1,2,3,4)

Let first root be mapped to 0, II to 1, iii to 3,...iv to 4th root

f(u*v) = f(cos 2pi/5+isin2pi/5)(cos4pi/5+isin4pi/5) (say)

Use Euler formula to get

f(u*v) = f(uv) = f(u)f(v)

Hence G is isomorphic to Z5 under addition.