Corollary 1

The exponent of a central simple algebra of finite degree over a commutative field divides its index.

It suffices to prove this for a field D of finite degree over its center K. Let L be a maximal commutative subfield of D that is a separable extension of K; we have $[D : K] = [L : K]^2$ (VIII, p. 265, Corollary 2, b) and c)). Then the extension L of K is a splitting field for the algebra D (VIII, p. 281, Proposition 5), and [L : K] coincides with the reduced degree of D. We then apply Theorem 6.