monic irreducible polynomials

Question

I need help with :

Let $\Bbb F_3=\Bbb Z_3$ be the field with 3 elements.Show that there are infinitely many monic irreducible polynomials in $\Bbb F_3[x]$ such that $P(0)=-1$.

Now,I saw this proof Ring of polynomials over a field has infinitely many primes but I am not sure,is there any different for this field and with the condition that $P(0)=-1$?

Answer 1

Given finitely many prime polynomials $p_1,p_2,\dots,p_n\in\mathbb F_3[x]$, consider that $xp_1(x)p_2(x)\cdots p_n(x)-1$.