Corollary 2

Let $N$ be a Galois extension of $K$. Suppose that the Galois group $\Gamma$ of $N$ over $K$ is the direct product of two closed subgroups $\Gamma_1$ and $\Gamma_2$ and denote by $E_i$ the field of invariants of $\Gamma_i$ for $i = 1, 2$. Then the canonical homomorphism of $E_1 \otimes_K E_2$ into $N$ is an isomorphism.

We have $E_1 \cap E_2 = K$ and $N = K(E_1 \cup E_2)$ by Cor. 6 (V, p. 69), and so it is enough to apply Theorem 5.