Show that $f(t) = t^2 + 4t + 2$ is irreducible over $\mathbb{F_5}$

Question

I had a very short introduction to some exercises about finite fields, but I don't understand the theory very well, so I'm a bit confused.

I have this polynomial $t^2 + 4t + 2$ with coefficients that are elements of $\mathbb{F_5}$. I wanted to try a "brute force method" to proof irreducibility over $\mathbb{F_5}$. I calculated the discriminant of the polynomial and I wanted to conclude that there exists no $g(t),h(t)$ with $\deg(g(t)),\deg(h(t)) < 2$ such that $f(t) = g(t)h(t)$. Is this a good way to proof this? And do I need to do all necessary calculations mod $5$?

A polynomial of degree $2$ or $3$ over a field is reducible if and only if it has a root in this field. In this case, you can just check directly if it has a root in $\mathbb{F_5}$ or not. Just substitute all $5$ elements. — Mark, May 10 '24 at 10:32
Did you know that the quadratic formula works just as well over any field (except characteristic 2)? You just have to reinterpret what fractions and square roots mean. — Arthur, May 10 '24 at 10:33
It may be easier to check that $(t+2)^2-2$ has no zeros in $\Bbb{F}_5$. Yes, you want to do the calculations modulo $5$, it's the whole point. — Jyrki Lahtonen, May 10 '24 at 10:50
But surely we have covered if not exactly this question, then very similar ones. Did you search? — Jyrki Lahtonen, May 10 '24 at 10:51
@Arthur No, I wasn't aware of that but I'm curious now. I want to look into it. Thank you for your comment. — user33, May 10 '24 at 10:55
@JyrkiLahtonen I was looking at similar questions, but apparently not similar enough, there seem to be multiple ways to check irreducibility and they seemed a bit more technical. — user33, May 10 '24 at 10:57
With a quadratic modulo a prime $p>2$ all you need to check is whether the discriminant is a quadratic residue. When the degree goes up, it gets progressively more taxing (even though there are algorithms faster than those checking for primality). — Jyrki Lahtonen, May 10 '24 at 10:58

Show that $f(t) = t^2 + 4t + 2$ is irreducible over $\mathbb{F_5}$

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