Remark 1

Here is another proof of Theorem 2. Let D be a finite field, let K be its center, and let L be a maximal commutative subfield of D. Let $x$ be an element of $D^*$. It belongs to a maximal commutative subfield $L_1$ of D. We have the equality

by Corollary 2 of VIII, p. 265, and therefore $[L : K] = [L_1: K]$. By Proposition 3 of V, §12, No. 2, p. 94, the extensions L and $L_1$ of K are isomorphic. By VIII, p. 263, Corollary, there exists an element $a$ of $D^*$ such that $aLa^{-1}= L_1$, so $a^{-1}xa$ belongs to L. We then have $(ay)^{-1}x(ay) =a^{-1}xa$ for every $y\in L^*$. Consequently, if S is a set of representatives of the left cosets of $D^*$ modulo $L^*$, then every element of $D^*-\{1\}$ can be written as $sxs^{-1}$ with $s\in S$ and $x\in L^*-\{1\}$. We denote the order of $D^*$ by $d$ and that of $L^*$ by $\ell$. Since the cardinal of S is equal to $d/\ell$, we have $d-1\leqslant (d/\ell )(\ell -1) =d-d/\ell$. It follows that $\ell =d$, and therefore L = D, which proves that the field D is commutative.