Showing existence of irreducible polynomial of degree 3 in $\mathbb{F}_p$ I'am trying to show that for every p$ \in \mathbb{N}$ where p is prime,
there is an irreducible polynomial of degree 3 in $\mathbb{F}_p$.
I've found too general answers for that question, but I want to show it in the most simple way.
I know to do so for polynomial of degree 2:
the function $x\mapsto x^2$ is not surjective, thus there is $a\in \mathbb{F}_p$ with $\forall b \in \mathbb{F}_p $ $b^2 - a \neq 0$
, which means  $x^2-a$ has no roots.
But I can't do the reduction to my problem.
Thanks.

If a degree $3$ polynomial is reducible over a field, then it has a root.  So you (just) need a degree $3$ polynomial without a root.
There are $\frac{p^3-p}3$ monic irreducible polynomials of degree $3$, according to this argument.
