Normal basis of an extension of degree 4 over its prime field.

Question

I want to construct a normal basis of $\mathbb{F}_{p^4}$ over $\mathbb{F}_p$, where $p$ is an odd prime. Is there any particular method to do it?

Answer 1

You have the chain $\Bbb F_p\subset\Bbb F_{p^2}\subset\Bbb F_{p^4}$. In particular, any one, say $\alpha$, of the $p^4-p^2$ elements not in the quadratic extension will have four conjugates. Then $\{\alpha,\alpha^p,\alpha^{p^2},\alpha^{p^3}\}$ is your normal basis, since the Galois group is always generated by Frobenius.