Theorem 2 (Wedderburn)

Every finite field is commutative.

Let D be a finite field, and let K be its center. The K-algebra D is central simple of degree $m^2$, where $m$ is a strictly positive integer. The reduced norm is a homogeneous polynomial mapping Nrd$: D\rightarrow K$ of degree $m$ (VIII, p. 345, Proposition 6), and we have Nrd($a$)$\not= 0$ for every $a\not= 0$ in D (VIII, p. 340, Proposition 3). The corollary above implies that $m\geqslant m^2$, and therefore $m= 1$. So we have D = K.