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..?