Can one use Eisenstein criteria in $\mathbb{F}_p[x]$

Question

Since I can not find this in a book, I want to ask the question here. Given a polynomial in a polynomial ring in one variable over $\mathbb{F}_p,$ can one use the Eisenstein criteria to decide if the polynomial is irreducible. For example we may take $x^7 + 6x^3 + 12x − 3$ which is irreducible in $\mathbb{Q}[x]$ but how about if we consider this in $\mathbb{F}_5 [x]$ ? Thanks.

Answer 1

$\mathbb{F_p}$ is a field, and so it doesn't have any irreducible elements. So Eisenstein criterion doesn't make sense here. And no, irreducibility over $\mathbb{Z}$ doesn't imply irreducibility over $\mathbb{F_p}$ after reducing the coefficients mod $p$. For example, $x^2+1$ splits in $\mathbb{F_2}[x]$.

Eisenstein criterion is useful when you want to check irreducibility of a polynomial in $R[x]$ over the field of fractions of $R$, where $R$ is some UFD domain. The classical cases are $R=\mathbb{Z}$ and $R=F[x]$ where $F$ is some field. So if you want to check irreducibility of a polynomial in two variables over $\mathbb{F_p}$, then you can indeed use the criterion. For example:

$t^p-x\in\mathbb{F_p}[x][t]$

is irreducible over $\mathbb{F}_p(x)$ by Eisenstein criterion, because $x$ is an irreducible element in $\mathbb{F_p}[x]$.