Lemma 1

Let V be a right vector K-space with a K'-structure V' and W a vector sub-K-space of V. There exist two vector sub-K-spaces $W_1, W_2$ of V, rational over K', such that V is the direct sum of $W_1$ and $W_2$ and, if V is identified with $W_1 \times W_2$, W is the graph of a K-linear mapping g of $W_1$ into $W_2$.

Let B be a basis of V rational over K'. Applying Theorem 2 of § 7, no. 1, to a basis of W over K, considered as a free subset of V, and the generating system the union of this free subset and B, it is seen that there exists a subset C of B such that V is the direct sum of W and the subspace $W_2$ of V generated by C. Also let $W_1$ be the subspace of V generated by $B - C$. As $B \subset V'$, clearly $W_1$ and $W_2$ are rational over K'. Moreover, for all $x \in W_1$, there exists one and only one vector $g(x)$ of $W_2$ such that $x + g(x) \in W$, since V is the direct sum of W and $W_2$; then W is the graph of g and g is K-linear since W is a vector sub-K-space of V.

Having proved this lemma, it is known that W is rational over a subfield L of K containing K' if and only if g is rational over L (no. 3, Proposition 4). The smallest field $K'(g)$ such that g is rational over $K'(g)$ is therefore also the smallest field over which W is rational.