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Number Theory - Primitive Roots

  1. Mar 22, 2007 #1
    Here is the question from the book:
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    Determine a primitive root modulo 19, and use it to find all the primitive roots.
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    [tex]\varphi(19)= 18[/tex]

    And 18 is the order of 2 modulo 19, so 2 is a primitive root modulo 19, but I am not sure of how to use that to find all primitive roots modulo 19. My only idea is that we need to find what values of g satisfy [itex]g^{18} \equiv 1 \ \text{mod 19}[/itex]. However, I am not sure how to solve that equation. Any ideas? Thanks!
     
  2. jcsd
  3. Mar 22, 2007 #2

    Hurkyl

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    Well, you know that the unit group of Z/19Z is simply a cyclic group of order 18, right?

    If that doesn't help, don't forget that g is a power of 2. Now, you know that everything in Z/19Z satisfies g^18 = 1... the things you're interested in are the things that do not also satisfy g^9 = 1 or g^6 = 1. (Do you see why?)
     
  4. Mar 22, 2007 #3
    Unfortunately I don't know much algebra, and our number theory class has not focused on the algebra behind it, so I don't really understand what you are saying.
     
  5. Mar 22, 2007 #4

    Hurkyl

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    Z/19Z is simply the residue classes modulo 19.
     
  6. Mar 24, 2007 #5
    Thanks, now I see the idea behind it, and I see how to find the others. Seems kind of obvious now :redface:
     
  7. Oct 24, 2010 #6
    why are we interested in the things that DO not satisfy g^9 = 1 or g^6 = 1?
     
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