Finite fields as roots of a Frobenius polynomial (${\mathbb{F}}_{p^n}=\{x:x^{p^n}=x\}$)

In a splitting field of $x^{p^n}-x$ over a prime field, the polynomial roots form a field. Frobenius identities close the polynomial roots under addition and multiplication, and nonzero polynomial roots have inverses which are again polynomial roots. The derivative is minus one, so the polynomial has exactly $p^n$ distinct polynomial roots. This constructs a field of every prime-power order.