I'm trying to find the orbits of the Frobenius homomorphism $\phi_p: a \to a^p$ on $\mathbb{F}_{p^4} / \mathbb{F}_p$. I can see there are $p$ 1 orbits corresponding to the action on $\mathbb{F}_p$ but I think there are also some order 2 and 4 orbits but I'm having trouble finding how many of each there are.
This is part of an argument on trying to show how many irreducible polynomials of degree 4 there are over $\mathbb{F}_p$ because I think this number should correspond to how many order 4 orbits there are, as then the polynomial with roots in that orbit should be irreducible.
Thanks