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 $\mathbb{Z}_{17}$.

You’re looking for a fourth root of $5$ in a field of characteristic $17$. You can show by direct computation that $5$ is a “primitive residue modulo $17$”, that is, that $5$ generates the multiplicative group $\Bbb F_{17}^*$ of nonzero elements of this finite field. In particular, it is not a fourth power, so that $X^4-5$ has no root in $\Bbb F_{17}$. This is not good enough.
But notice that $5$ has period $16$ in $\Bbb F_{17}^*$, and its fourth root must have period $4\cdot16=64$ in its field. What is the smallest field of characteristic $17$ that has elements of period $64$? We need $17^n-1$ to be divisible by $64$, and the smallest such number is $n=4$, for $(17^2-1)/64=9/2$, but $17^4-1=64\cdot1305$. Thus the minimal polynomial for $\sqrt[4]5$ over $\Bbb F_{17}$ must be $X^4-5$ : an irreducible polynomial.
