Exercise 5, § 8

Let $K_0$ be a subfield of a field K such that K is of dimension 2 as a right vector space over $K_0$. Let E be a left vector space over K.

(a) Let $E_0$ be a subset of E which is a vector space over $K_0$ and let V be the largest vector sub-K-space of $E_0$; if $W_0$ is a vector sub-$K_0$-space of $E_0$ supplementary to V, show that the vector sub-K-space W of E generated by $W_0$ is such that $V \cap W = \{0\}$ (note that if an element $\mu \in K$ does not belong to $K_0$, the relation $\mu x \in E_0$ for an $x \in E_0$ implies $x \in V$).

(b) Let $E'_0$ be another vector sub-$K_0$-space of E and $V'$ the largest vector sub-K-space of $E'_0$. For there to exist a K-automorphism of E mapping $E_0$ to $E'_0$, it is necessary and sufficient that V and $V'$ have the same dimension with respect to K, that the codimensions of V in $E_0$ and $V'$ in $E'_0$ (with respect to $K_0$) be equal and that the codimensions of $E_0$ and $E'_0$ in E (with respect to $K_0$) be equal (use (a)).