Adjoining Elements to Finite Fields: Finding Generators

[R.P.F.]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I did the F_5(2^{1/2}) case and think the generator is 2+\sqrt{2}. But don't know how this generalizes..
Thanks!

 

[micromass]

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

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.

 

[R.P.F.]

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

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.

Really? Even the fact that \mathbb{F}_5 itself is cyclic does not help..?