How to formally prove that a degree 2 or 3 polynomial f(x) ∈ F[x] is irreducible if and only if it has no root in F

Question

I am trying to prove that $ F_2[x]/(x^3 + x + 1)$ is a field, but that $F_3[x]/(x ^3 + x + 1)$ is not a field.

A solution here uses a lemma:

A degree 2 or 3 polynomial $f(x) ∈ F[x]$ is irreducible if and only if it has no linear factor $x − a ∈ F[x]$, i.e., no root in $F$.

But could someone tell me how to prove it? And how does it imply the fact that there is no root in $F$?

I am stuck at proving that it must have a root in F.

Thank in advance!

Answer 1

Suppose a polynomial $f$ of degree $2$ or $3$ were reducible; i.e. suppose we have $f(x) = g(x)h(x)$ where $g$ and $h$ are nonconstant polynomials. This means $\deg(g)$ and $\deg(h)$ are necessarily $\geq 1$. How is $\deg(f)$ related to $\deg(g)$ and $\deg(h)$ given that we are working over a field? What can we conclude?