1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Multiplying primitive roots of unity.

  1. Jul 30, 2013 #1
    1. The problem statement, all variables and given/known data


    Let ##ζ_3## and ##ζ_5## denote the 3rd and 5th primitive roots of unity respectively. I was wondering if I could write the product of these in the form ##ζ_n^k## for some n and k.


    2. Relevant equations



    3. The attempt at a solution



    We know that ##ζ_3## is a root of ##x^3=1##, and ##ζ_5## is a root of ##x^5=1##, so ##ζ_3ζ_5## must be a root of ##x^{15}=1##, right?...so ##ζ_3ζ_5 = ζ_{15}##. And in general ##ζ_nζ_k = ζ_{[n,k]}##, where [n,k] denotes the lcm of n and k. Is that right?

    Also is it true that ##ζ_{15}^8 = ζ_{15}##? If so, then why/how?

    Thank you in advance
     
  2. jcsd
  3. Jul 30, 2013 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You got to be careful to specify which root you work with. It is certainly true that ##\zeta_{15}^8## is a primitive 15th root of unity, but it might not be the same one!

    For example, in ##\mathbb{C}##, we have ##\zeta_{15} = e^{2\pi i /15}## is a possible choice. But then ##\zeta_{15}^8 = e^{2\pi i (8/15)}## is a 15th root of unity but not the same one.

    It is indeed true that ##\zeta_3\zeta_5## is a primitive 15th root of unity. And depending on how you defined ##\zeta_{15}##, equality holds. For example:

    [tex]e^{2\pi i/3} e^{2\pi i/5} = e^{ 2\pi i (8/15) }[/tex]

    So this is a primitive 15th root of unity. But it is not ##\zeta_{15}## if you defined ##\zeta_{15} = e^{2\pi i /15}##.
     
  4. Jul 30, 2013 #3
    Perhaps instead of using [itex] ζ_{n} [/itex] to represent a particular primitive n-th root, you might use it to denote the whole set of primitive n-th roots. Then change "=" with [itex] \in [/itex], and define the product of two sets as the set containing all possible products of elements from either set. Then what you say will be true.
     
    Last edited: Jul 30, 2013
  5. Jul 30, 2013 #4
    Thanks. We can also conclude that ##\Bbb{Q}(\zeta_{15}^8) = \Bbb{Q}(\zeta_{15})## since both are primitive roots of unity and ##\zeta_{15}^{8k}=\zeta_{15}## and ##\zeta_{15}^m = \zeta_{15}^8## for some ##k## and ##m##, right?
     
  6. Jul 30, 2013 #5

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Sure.
     
  7. Jul 30, 2013 #6

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    But note that it doesn't work for ##\zeta_{15}^5##, for example.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Multiplying primitive roots of unity.
  1. Roots of unity (Replies: 1)

  2. Roots of unity (Replies: 9)

Loading...