# Adjoining element to a field

1. Oct 22, 2011

### R.P.F.

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

3. 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!

2. Oct 22, 2011

### micromass

Staff Emeritus
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.

3. Oct 22, 2011

### R.P.F.

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