Proposition 8

— The N-linear mapping $\psi$ of $N \otimes_K M$ into $V = \prod_{\sigma \in \Gamma} M^\sigma$ is injective, and it is bijective if $\Gamma$ is finite.

For every $a \in \Gamma$ we define a mapping $u_\sigma$ of $V$ into $V$ by

for $h \in V$ and $\tau \in \Gamma$. The verification of (1)-(4) is immediate. Denote by $V$, the set of $h \in V$ such that $u_\sigma(h) = h$ for all $\sigma \in \Gamma$. For each $x \in M$ let $\theta(x)$ be the constant mapping of $\Gamma$ into $M$ with value $x$; then $\theta$ is a K-isomorphism of $M$ onto $V$. If we define the homomorphism $\varphi : N \otimes_K V, \to V$ as above, we have $\psi = \varphi \circ (\mathrm{Id}_N \otimes \theta)$ and now Prop. 8 follows from Prop. 7.