TITLE: For which of the following fields $\mathbb F$ the polynomial $x^3-312312x+123123$ is irreducible in $\mathbb F[x]$?
QUESTION [6 upvotes]: I came across the following problem:  

For which of the following fields $\mathbb F$ the polynomial  $x^3-312312x+123123$ is irreducible in $\mathbb F[x]$? The options are
  $1.$ the field $\mathbb F_3$ with $3$ elements.
  $2.$ the field $\mathbb F_7$ with $7$ elements.
  $3.$ the field $\mathbb F_{13}$ with $13$ elements.
  $4.$ the field $\mathbb Q$ of rational numbers.  

Can someone point me in the right direction with some explanation? Thanks in advance for your time.

REPLY [2 votes]: If we take field $\mathbb F_3$, then the polynomial become $x^3$ (since other coefficients are divisible by $3$). Same for $\mathbb F_7$ and $\mathbb F_{13}$, and $x^3$ is not irreducible. So only $(d)$ is correct.