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Math Help - Primitive roots

  1. #1
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    Primitive roots

    I am trying to show that the primitive fifth root of unity e^(2(pi)i/5)=sqrt5-1/4 + i (sqrt (5)+(sqrt5)/8. I have solved (sqrt (5)-sqrt5/8 but not sure really how to complete the algebra?
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    I am trying to show that the primitive fifth root of unity
    We know the n>1-th roots of unity that are of the form e^{m*pi*i/n} where m is a positive integer and gcd(m,n)=1 will be a generator for the group of all roots of unity.
    Since 5 is a prime there are exactly 4 primitive roots of unities (because those the ones that form generators):
    e^{pi*i/5},e^{2pi*i/5},e^{3pi*i/6},e^{4pi*i/5}
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  3. #3
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    Sorry but

    Quote Originally Posted by chadlyter View Post
    I am trying to show that the primitive fifth root of unity e^(2(pi)i/5)=sqrt5-1/4 + i (sqrt (5)+(sqrt5)/8. I have solved (sqrt (5)-sqrt5/8 but not sure really how to complete the algebra?
    I'm sorry I can't answer your question but I bet you can answer mine. What is a primitive root?
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  4. #4
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    Hello, chadlyter!

    More careful placement of parentheses would have helped.

    And I still don't understand what you did . . .


    I am trying to show that the primitive fifth root of unity is:
    . . . . . - - - . . . . . ._ . . . . . . . . _________
    . . e^{2πi/5} .= .[√5 - 1]/4 + i√(5 + √5)/8

    . . . . . . . . . . . ________
    I have solved √(5 - √5)/8 . ? . and what do you mean by "solved"?

    From DeMoivre's Formula, we already know that:

    . . e^(2πi/5) .= .cos(2π/5) + isin(2π/5)


    So we need to find the values of: .cos(72) and sin(72)

    Can you do that?

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  5. #5
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    Quote Originally Posted by exweedfarmer View Post
    I'm sorry I can't answer your question but I bet you can answer mine. What is a primitive root?
    It is not an elementary answer but here it is.


    Given a finite field F. It can be shown that the multiplicative group is in fact cyclic! Thus, it has generators. Those generators are primitive roots.
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