Proposition 11

Let $G_1$ and $G_2$ be groups. Let $L_1$ and $L_2$ be Galois algebras over K with respective actions $\lambda_1: G_1\rightarrow$ Aut$_K(L_1)$ and $\lambda_2: G_2\rightarrow$ Aut$_K(L_2)$. Set $L = L_1\otimes_KL_2,G = G_1\times G_2$, and $\lambda (g_1, g_2) =\lambda_1(g_1)\otimes \lambda_2(g_2)$ for $(g_1, g_2)\in G$. Then the K-algebra L endowed with the action $\lambda$ is a Galois algebra over K.

We reason as before, taking the following into account: If $L_1$ and $L_2$ are étale, then so is the algebra $L = L_1\otimes_KL_2$ (V, §6, No. 5, p. 32, Corollary 1), and we have the equalities

$[L : K] = [L_1: K][L_2: K]$ and Card(G) = Card(G$_1$) Card(G$_2$).

Moreover, if $L^{G_i}_i$ denotes the ring of invariants of $G_i$ in $L_i$, then it follows from Lemma 11 below that $L^{G_1}_1\otimes_KL^{G_2}_2$ is the ring of invariants of $G_1\times G_2$ in $L_1\otimes_KL_2$.