Exercise 3, § 8

Let K be a field, L a subfield of K and let $K_L$ denote the field K considered as a left vector space over L; $E = \mathrm{End}_L(K_L)$ has canonically a (K, K)-bimodule structure. The dual $(K_L)^*$ is contained in E and is a (K, L)-sub-bimodule of E. When L is contained in the centre of K, the two vector L-space structures on E, obtained by restricting to L the field of scalars of the two vector K-space structures on E, are identical.

Suppose henceforth that L is contained in the centre of K and that the dimension of $K_L$ is finite and equal to n.

(a) Show that $(K_L)^*$ is 1-dimensional under its left vector space structure over K (calculate the dimension of $(K_L)^*$ over L in two ways).

(b) Show that E is an n-dimensional left vector space over K (same method).

(c) Let F be a finite-dimensional left vector space over K and $F_{[L]}$ the corresponding vector space over L. If $u_0 \neq 0$ is a linear form on $K_L$, show that the mapping $x' \mapsto u_0 \circ x'$ of the dual $F^*$ of F (considered as a vector space over L) onto the dual $(F_{[L]})^*$ of $F_{[L]}$ is bijective (use (a)).