Irreducible polynomial of degree 7 in $\mathbb{Z}_5[x]$

Question

I'm looking for an example of a monic degree 7 polynomial that is irreducible over $\mathbb{Z}_5$, or suggestions for how to construct one (searching on Google for examples of such polynomials did not provide any answers).

Answer 1

Given a prime power $q$ and a positive integer $k$, there is exactly one extension (up to isomorphism) of $\Bbb F_q$ of degree $k$, namely $\Bbb F_{q^k}$. Every element of $\Bbb F_{q^k}$ is a root of the polynomial $x^{q^k} - x$ (this is a consequence of the fact that the multiplicative group of a finite field is cyclic). In particular, every element $\alpha$ such that $\Bbb F_{q^k} = \Bbb F_q[\alpha]$ is the root of some degree-$k$ polynomial that is irreducible over $\Bbb F_q$, so that irreducible polynomial must divide $x^{q^k}-x$.

In this particular case $q=5$ and $k=7$, then, you can actually find every single (monic) irreducible polynomial of degree $7$ over $\Bbb F_5$ by "simply" factoring $x^{5^7}-x$ and keeping only the degree-7 factors. Fortunately, computers can do this: try going to the Wolfram Cloud and evaluating

Factor[x^(5^7) - x, Modulus -> 5]

to see all 11,160 such polynomials (together with the expected 5 linear polynomials) dividing $x^{5^7}-x$.