Exercise 6, § 14

Let D be a field with center K, distinct from K, such that every element of D is algebraic over K.

a) Prove that D contains a separable commutative extension of degree $>1$.

b) Let $m$ be an integer; suppose that every element of D has degree $\leqslant m$ over K. Prove that we have $[D : K]\leqslant m^2$ (first prove that D has finite degree using a) and Theorem 5 of VIII, p. 259).