Exercise 4, § 8

Let K be a field of finite rank over its centre Z and L a subfield of K containing Z.

(a) Show that the left and right vector space structures of K with respect to L have the same dimension.

(b) Show that properties (a), (b), (c) of Exercise 3 also hold in this case (if $u_0$ is a linear form $\neq 0$ on $L_Z$ and $v_0$ a linear form $\neq 0$ on $K_L$, note that $\xi \cdot (u_0 \circ v_0) = u_0 \circ (\xi v_0)$ for $\xi \in K$, that $\xi \cdot (u_0 \circ v_0)$ runs through the dual $(K_Z)^*$ when $\xi$ runs through K and use Exercise 3 (c)).