Prime ideals of $\mathbb{F}_p[X]$

Question

I want prove that every prime ideal of $\mathbb{F}_p[X]$ is of the form $(\overline{P(X)})$, where $P(X)\in\mathbb{Z}[X]$ is irreducible. Any help?

Answer 1

In fact every non-zero ideal (prime or not) in $\mathbb F_p[X]$ is of the form $(\overline{P(X)})$ where $P(X) \in \mathbb Z[X]$ is irreducible.

We are in a PID, hence any ideal is generated by one single polynomial and you can always choose an irreducible representative in $\mathbb Z[X]$, because you can change the coefficients by multiples of $p$ and thus you can choose the representative to be $q$-Eisenstein for a fixed prime number $q \neq p$.