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(Roots of Unity) 𝜁 k = exp((2𝜋k i)/n) How to show that 𝜁 k is a root of unity for each k= 1,2,3...n. I know that in order for 𝜁 k to be a root of unity it needs to equal 1, so do I substitute k values and check if they equal 1? I don't know how to begin solving this question. Please help. Thank you.

(Roots of Unity)
𝜁k= exp((2𝜋ki)/n)
How to show that 𝜁is a root of unity for each k= 1,2,3...n.
 
I know that in order for 𝜁k to be a root of unity it needs to equal 1, so do I substitute k values and check if they equal 1? I don't know how to begin solving this question. Please help. Thank you. 

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
4 years ago
this is very easy to explain. you have asked how to show that 𝜁is a root of unity for each k= 1,2,3...n.
if 𝜁is a root, then 𝜁k^n should be equal to 1.
or [exp((2𝜋ki)/n)]^n= exp((2𝜋ki)/n)*n= exp(i2𝜋k)= cos(2𝜋k)+i*sin(2𝜋k)= 1+0= 1
thus we have proved that it is 1 and hence 𝜁is an nth root of 1 for all integral k.

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