Irreducibility of $x^4-5$ over $\mathbb{Z}_{17}$ Obviously, it'd be hard to try all the $17$ elements to see if there is some root, and even if there is none, it'd be necessary to verify if it can't be factored into two irreducible 2 degree polynomials. The Eisenstein criterion will just be able to say about irreducibility on $\mathbb{Q}[x]$.
What can I do?
I must also verify it for the polynomials $x^3-5$ and $x^4+7$, both over $\mathbb{Z}_{17}$.

If $x^4\equiv 5\pmod{17}$, then $x^{16}\equiv 5^4\equiv 13\pmod{17}$, but this contradicts Fermat's Little theorem.
More strongly, $y^2\equiv 5\pmod{17}$ has no solution. By Quadratic Reciprocity:
$$\left(\frac{5}{17}\right)=\left(\frac{17}{5}\right)=\left(\frac{2}{5}\right)=-1$$
