I got a couple questions I'm trying to work out.
a) Show that $x^4+2$ is irreducible in $\mathbb F_5[x]$
2) Describe the quoteint $\mathbb F_5[x]/(x^4+2)$
3) Show that $f(x) = x^4+5x^2-3$ is irreducible in $\mathbb Q[x]$.
For a), do I just try to come up with quadratic factors and show some sort of a conclusion?
b), I know that it will be a field of $5^4$ elements assuming that result from a) is true. Is there any other properties that I should note?
c) I thought if I could show that $f(x)$ is irreducible in $\mathbb Z[x]$, then by Gauss' Lemma, I can say that it is irreducible in $\mathbb Q[x]$ since the function is monic. But not sure how to show the irreducibility in $\mathbb Z[x]$, besides just trying to brute force with quadratic factors.
Any help would e appreciated!