Corollary 2

Let D be a field of finite degree over K with center K, and let L be an extension of K of finite degree that is a splitting field for D. The reduced degree of D divides [L : K].

Denote the reduced degree of D (VIII, p. 253) by $r$; by definition, we have $[D : K] =r^2$. By Proposition 5, there exists a central simple K-algebra B similar to D of which L is a maximal commutative subalgebra. Since B is isomorphic to a matrix algebra $\mathbf{M}_n(D)$ (VIII, p. 278, Lemma 1), we have $[B : K] =n^2r^2$ and consequently $[L : K] =nr$ (VIII, p. 262, Proposition 3).