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Category: Calculus and Above
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Experience:  B.S. in Civil Engineering
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# I am trying to prove isomorphism between a fifth root of

### Customer Question

I am trying to prove isomorphism between a fifth root of unity and z sub 5 under addition. I know I have to show one to one, onto, and homomorphism. But I can't seem to get started. I've already proven that the fifth root of unity is a group, using de moivre's formula, but I can't seem to wrap my head around how to map between the two groups to find my one-to-one.
Submitted: 1 year ago.
Category: Calculus and Above
Expert:  Ryan replied 1 year ago.

Hi,

Thank you for using the site. I'll be happy to help you with this.

I've seen this problem several times, and the mapping function that I have used is:

φ(n) = cos(2n*pi/5) + i * sin(2n*pi/5)

This function will map the values in Z5 to the group of fifth roots that you got from DeMoivre's formula.

A list of the calculated values of φ(n) for all n in Z5 (that is, n = 0, 1, 2, 3, 4) will show that φ(n) is both one-to-one and onto.

Let me know if you need more than this. I can send the complete solution for this part if you need it.

Thanks,

Ryan

Customer: replied 1 year ago.
Unfortunately, I will need to see more. I kind of know that equation already, since it's the derivation of de Moivre's formula for the fifth roots of unity. But the actual problem I'm having is how to use it to map from the group to the Z5 under addition. So I guess I need to see how you plug one in to get the other?
Expert:  Ryan replied 1 year ago.

No problem. Here is the complete solution for both parts of this problem:

DeMoivre