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Homework Help: 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..
  2. jcsd
  3. Oct 22, 2011 #2
    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..?
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