Proposition 7

— Let $V$ be a vector space over $N$, $(u_\sigma)_\sigma$, r a family of mappings of $V$ into itself satisfying (1) to (4) and $V_0$ the set of $x \in V$ such that $u_\sigma(x) = x$ for all $\sigma \in \Gamma$.

a) $V_0$ is a vector sub-K-space of $V$ and the K-linear mapping $\varphi$ of $N \otimes_K V_0$ into $V$ which maps $\lambda \otimes x$ to $\lambda x$ is injective.

b) If $\Gamma$ is finite, then $\varphi$ is bijective and $V_0$ is a K-structure on $V$.

It is clear that $V_0$ is a vector sub-K-space of $V$.

The formula $u_\sigma \circ \varphi = \varphi \circ (\sigma \otimes \mathrm{Id}_{V_0})$ shows that the kernel $W$ of $\varphi$ is stable under the mappings $\sigma \otimes \mathrm{Id}_{V_0}$; by the Cor. to Prop. 6 there exists therefore a subspace $W_0$ of $V_0$ such that $W = N \otimes_K W_0$. If $x$ belongs to $W_0$ we then have $x = \varphi(1 \otimes x) = 0$, hence $W_0 = 0$ and so $W = 0$. This proves a).

Suppose now that $\Gamma$ is finite; we have to show that $\varphi$ is surjective, or equivalently that $V_0$ generates the vector N-space $V$. Thus let $f$ be an N-linear form on $V$ whose restriction to $V_0$ is zero. Let $x \in V$; for every $\lambda \in N$ the element $y_\lambda = \sum_{\sigma \in \Gamma} u_\sigma(\lambda x)$ of $V$ clearly belongs to $V_0$, whence $f(y_\lambda) = 0$, that is,

By Dedekind's theorem (V, p. 27, Cor. 2) we thus have $f(u_\sigma(x)) = 0$ for each $\sigma \in \Gamma$; in particular, taking $\sigma = \varepsilon$ we find $f(x) = 0$, which means that $f = 0$. This proves b).

Let $M$ be a vector space over $N$; for each $\sigma \in \Gamma$ let $M^\sigma$ be the vector space over $N$ with the same underlying additive group as $M$, with the external law $(A, x) \mapsto \sigma(\lambda) x$. Write $V = \prod_{\sigma \in \Gamma} M^\sigma$; the underlying additive group of $V$ is that of all mappings of $\Gamma$ into $M$, with the external law defined by

(The product $\sigma(\lambda) h(\sigma)$ is calculated in the vector space $M$.) Further, we define on $N \otimes_K M$ a vector space structure over $N$ by the formula Finally we denote by $\psi$ the K-linear mapping of $N \otimes_K M$ into $V$ characterized by the relation for $A \in N, x \in M$ and $\sigma \in \Gamma$. It is clear that $\psi$ is N-linear