Can we have two different polynomials of the same degree $d$ here in the factorisation of $x^{p^n} -x$?

Question

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 $x^{p^n} -x$?

Answer 1

Yes, of course. One simple example: over $\mathbb{F}_2$, $$x^{2^3}-x=x(x+1)(x^3+x^2+1)(x^3+x+1)$$

Answer 2

The simplest example: take $p=2,n=1$, and you get $$ x^2-x=x(x-1) $$

Answer 3

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).