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!

Adjoining element to a field

  1. Oct 22, 2011 #1
    1. The problem statement, all variables and given/known data

    I have the field [tex] F_5 [/tex] and I adjoin some square root of 2 , say [tex]2^{1/4}[/tex]. Is there a way to see that the multiplicative group inside [tex]F_5(2^{1/4})[/tex] is cyclic and find the generator?

    2. Relevant equations



    3. The attempt at a solution

    I did the [tex]F_5(2^{1/2})[/tex] case and think the generator is [tex]2+\sqrt{2}[/tex]. But don't know how this generalizes..
    Thanks!
     
  2. jcsd
  3. Oct 22, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    I don't really like the exponent notation. You should write your ring as [itex]\mathbb{F}_5[X]/(X^4-2)[/itex].

    Finding a generator for the cyclic group is a quite difficult problem and still an active problem of research. I fear that the only solution is to test all the elements and see whether they are cyclic.
     
  4. Oct 22, 2011 #3
    Really?:cry::cry: Even the fact that [tex]\mathbb{F}_5[/tex] itself is cyclic does not help..?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook