In Hartshorne II.2.11, I am tasked with describing $\text{Spec}(\mathbb{F}_p[x])$. Now since $\mathbb{F}_p$ is a field, the corresponding polynomial ring is a UFD so the prime elements are exactly those that are irreducible. Since it is also an integral domain, I get that $\text{Spec}(\mathbb{F}_p[x])=(0)\cup\bigcup_i(f_i)$ where $f_i$ is irreducible and monic. My question is simple. Is there a nice algebraic way to describe $\bigcup_i (f_i)$?
I know that the algebraic closure of $\mathbb{F}_p$ is $\bigcup_{i\geq 1} \mathbb{F}_{p^i}$, and that the splitting field of $x^{p^n} - x$ is $\mathbb{F}_{p^n}$. Is there a way to describe $\bigcup_i (f_i)$ as some sort of limit for instance?