<s>[INST] Can we have two different polynomials of the same degree $d$ here in the factorisation of $x^{p^n} -x$? In the proposition "The polynomial $x^{p^n} -x$ is precisely the product of all the distinct irreducible polynomials in $\Bbb F_p[x]$ of degree $d$ where $d$ runs through all divisors of $n$."
Can we have two different polynomials of the same degree $d$ here in the factorisation of $x^{p^n} -x$?
 [/INST] Well, another example is
$$x^{16}-x = x(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1),$$
where the polynomials $x^4+x+1$ and $x^4+x^3+1$ are primitive and conjugate and the polynomial $x^4+x^3+x^2+x+1$ is not primitive (the roots are 5th roots of unity).
</s>