Theorem 1

Let $\mathscr{M}$ be a subset of $\mathrm{End}_{K'}(K)$ and $L = \chi(\mathscr{M})$ the subfield of K the centralizer of $\mathscr{M}$.

(i) Let V be a right vector K-space with a K'-structure. For a vector $x \in V$ to be rational over L, it is necessary and sufficient that $\phi_V(x.\eta) = x.\phi(\eta)$ for all $\phi \in \mathscr{M}$ and all $\eta \in K$. For a vector sub-K-space W of V to be rational over L, it is necessary and sufficient that $\phi_V(W) \subset W$ for all $\phi \in \mathscr{M}$.

(ii) Let $V_1, V_2$ be two right vector K-spaces each with a K'-structure. For a K-linear mapping f of $V_1$ into $V_2$ to be rational over L, it is necessary and sufficient that $f(\phi_{V_1}(x_1)) = \phi_{V_2}(f(x_1))$ for all $x_1 \in V_1$ and all $\phi \in \mathscr{M}$.

We first prove assertion (i) for x. Let B be a basis of V rational over K' and write $x = \sum_{b \in B} b.\xi_b$; then, for $\phi \in \mathscr{M}$ and $\eta \in K$,

and therefore, the relations

"for all $\phi \in \mathscr{M}$ and all $\eta \in K$, $\phi_V(x.\eta) = x.\phi(\eta)$"

and

"for all $\phi \in \mathscr{M}$, all $b \in B$ and all $\eta \in K$, $\phi(\xi_b\eta) = \xi_b.\phi(\eta)$"

are equivalent. The second of these relations means that for all $b \in B$, $\xi_b \in \chi(\mathscr{M})$, which proves the first assertion of (i).

We next prove (ii). For f to be rational over L, it is necessary and sufficient that, for all $x'_1 \in V_1$ rational over K', $f(x'_1)$ is a vector in $V_2$ rational over L; this will imply that $f(x_1)$ is rational over L for every vector $x_1$ of $V_1$ rational over L, such a vector being a linear combination with coefficients in L of vectors rational over K'. The above condition is equivalent, by the first part of the argument, to the relation

which may also be written

As every element of $V_1$ is a linear combination with coefficients in K of elements of $V_1$ rational over K', condition (5) is equivalent to

for all $x_1 \in V_1$ and all $\phi \in \mathscr{M}$.

Finally, to prove the second assertion in (i), we use no. 6, Lemma 1: W is the graph of a K-linear mapping $g : W_1 \to W_2$ and W is rational over if and only if the mapping g is rational over L (no. 3, Proposition 4). By (ii), for g to be rational over L, it is necessary and sufficient that $g(\phi_{W_1}(x_1)) = \phi_{W_2}(g(x_1))$ for all $x_1 \in W_1$ and all $\phi \in \mathscr{M}$; as $\phi_v = \phi_{W_1} \times \phi_{W_2}$, the above condition means that the graph W of g is stable under $\phi_v$ for all $\phi \in \mathscr{M}$.