$F_{p^n}$ can be realized as $F_p(\alpha)$ where $\alpha$ is generator of $F_{p^n}^{\times}$. Since $\alpha$ is a root of $x^{p^n} - x$, then the minimal polynomial of $\alpha$ divides $x^{p^n} - x$.
Suppose that $g(x)$ is irreducible polynomial polynomial of degree d dividing n. If $\eta$ is a root of g(x), then $F_p(\eta)$ is a subfield of $F_{p^n}$ of degree d. Since d | n so $F_p(\eta) = F_{p^d}$ so the roots of g(x) is $F_{p^n}$.
The elements of $F_p^n$ are roots of $x^{p^n} - x$. If we collect the factors of $x - \eta$ of this polynomial according to the degree of the minimal polyno…
I recently saw somewhere that
$$\sum_{n=1}^{\infty} (n) = -\frac {1}{12}$$
Now I do not want to go over the proof of that. I accept that the justification I saw was satisfactory for my taste (although incredibly mind boggling).
That got me thinking. I can rewrite that sum as the following:
$$...
