<s>[INST] Is the polynomial $6x^4+3x^3+6x^2+2x+5\in GF(7)[x]$ irreducible? Is the polynomial $6x^4+3x^3+6x^2+2x+5\in GF(7)[x]$ irreducible?
What is the best/simplest/elementary way to approach this? Any solutions or hints are greatly appreciated.
 [/INST] Because the modulus is so small, we can do this the hard way.
All arithmetic will be done in $\mathbb Z_7.$
$p(x)=6x^4+3x^3+6x^2+2x+5$
$-p(x) = x^4 + 4x^3 + x^2 + 5x + 2$
$-p(x-1) = x^4 + 2x^2 + 4x + 2$
Now we suppose that
\begin{align}
   -p(x-1)
      &= (x^2 + ax + b)(x^2 + cx + d)\\
      &= x^4 + (a+c)x^3 + (ac + b + d)x^2 + (ad + bc)x + bd
\end{align}
We end up with the equations
\begin{align}
   a+c &= 0\\
   ac + b + d &= 2\\
   ad+bc &= 4\\
   bd &= 2
\end{align}
Iterating through $b$, we finally get $(a,b,c,d) = (4,1,3,2)$.
So 
\begin{align}
   -p(x-1) &= (x^2 + 4x + 1)(x^2 + 3x + 2)\\
    p(x-1) &= -(x^2 + 4x + 1)(x^2 + 3x + 2)\\
      p(x) &= -(x^2 + 5x + 3)(x^2 + 6x + 3)\\
\end{align}
</s>