Prove the following polynomials are irreducible over $\mathbb Z_5[x]$:
$2x^3+x^2+4x+1$ and $x^4+1$.
So the exponents are already as reduced as they can be in terms of $\mathbb Z_5$. I cannot find any rational roots by factoring or using the rational root test. I don't believe I can't use Eisenstein's criterion on the first polynomial, the second polynomial concludes irreducibility over $\mathbb Q$. Can I then conclude that since no rational roots can be found that they are irreducible over $\mathbb Z_5$?
EDIT: Using N.S.'s advice I will use $a(x)=2x^3+x^2+4x+1$ where $x=0,-1,1,-2,2$ which all give numbers other than zero, thus there are no roots in $\mathbb Z_5[x]$.
Best Answer
Since you work over $\mathbb Z_5$, rational root test is useless.
For the first one, since it has degree 3, it is irreducible if and only if it doesn't have roots. And there are only 5 possible roots in $\mathbb Z_5$, try them.
For the second, note that $$X^4+1=X^4-4$$