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Math Help - Roots of unity isomorphic to cyclic group?

  1. #1
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    Roots of unity isomorphic to cyclic group?

    Hello there!

    I've been tasked with showing that A_n, where A_n = {z, exists in the complex numbers | z^n = 1 } is isomorphic to the cyclic group Cn, where Cn is of order n.

    My understanding (or memory, rather) of complex numbers is quite dim, so this problem is becoming quite of a hurdle.
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  2. #2
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    Quote Originally Posted by charleschafsky View Post
    Hello there!

    I've been tasked with showing that A_n, where A_n = {z, exists in the complex numbers | z^n = 1 } is isomorphic to the cyclic group Cn, where Cn is of order n.

    My understanding (or memory, rather) of complex numbers is quite dim, so this problem is becoming quite of a hurdle.
    Hint: The solutions to z^n = 1 are \zeta, \zeta^2, ... , \zeta^n where \zeta = e^{2\pi i/n}.
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  3. #3
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    First note that all the nth roots of unity are powers of e^{\frac{2 \pi i}{n}}
    Now define a function f:C_n \rightarrow \mathbb{C}
    where
    f(a)=e^{a\frac{2\pi i}{n}}

    Can you show f is an isomorphism?
    Last edited by whipflip15; November 24th 2008 at 12:16 AM.
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  4. #4
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    define an isomorphism from all roots of unity to An. I did this problem before, I forget how the isomorphism went. But it is not hard to show.
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