Proposition 6

— a) Let $V$ be a vector space over $N$ with a $K$-*structure*. For a vector $x \in V$ to be rational over $K$ it is necessary and sufficient that $u_\sigma(x) = x$ for all $\sigma \in \Gamma$. For a vector sub-N-space $W$ of $V$ to be rational over $K$ it is necessary and sufficient that $u_\sigma(W) \subset W$ for all $\sigma \in \Gamma$.

b) Let $V_1$ and $V_2$ be two vector spaces over $N$, each with a $K$-*structure*. For a linear mapping $f$ of $V_1$ into $V_2$ to be rational over $K$ it is necessary and sufficient that $f(u_\sigma(x)) = u_\sigma(f(x))$ for all $\sigma \in \Gamma$ and all $x \in V_1$.

It is clear that $K$ is the set of $x \in N$ such that $\sigma(xy) = x \sigma(y)$ for all $\sigma \in \Gamma$ and all $y \in N$. The proposition therefore follows from Th. 1 (II, p. 324).