Corollary 2

Let D be a field of finite degree over K with center K.

a) The maximal commutative subfields of D are the maximal commutative subalgebras of D and also the maximal semisimple commutative subalgebras of D. Every commutative subfield L of D is contained in a maximal commutative subfield.

b) Let L be a commutative subfield of D that is a separable extension of K; it is contained in a maximal commutative subfield of D that is a separable extension of K.

c) Let L be a commutative subfield of D. Then L is a maximal commutative subfield of D if and only if we have $[D : K] = [L : K]^2$.

A subalgebra of D is a field (V, §2, No. 2, p. 10, Proposition 1) and is therefore semisimple. Moreover, a maximal commutative subfield of D contains K. Assertions a) and b) then follow from Corollary 1 and assertion c) of Proposition 3 (VIII, p. 262).