= Finite fields as roots of a Frobenius polynomial
{title2=$\mathbb F_{p^n}=\{x:x^{p^n}=x\}$}
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.
Back to article page
