Corollary 1

Let $E'$ be a subfield of $N'$ containing $K'$. There exists a unique subfield $E$ of $N$ containing $K$ and such that $E' = K'(E)$. We have $E = E' \cap N$.

Put $E = E' \cap N$, then $E' \supset K'(E)$. Now put $\Gamma = \mathrm{Gal}(N/K)$ and $A = \mathrm{Gal}(N/E)$, and define $\Gamma'$ and $A'$ similarly. The mapping $A : a \mapsto a|_N$ is an isomorphism of $\Gamma'$ onto $\Gamma$ and also of $A'$ onto $A$; in other words, $A'$ consists of those $\sigma \in \Gamma'$ for which $\lambda(\sigma)$ belongs to $A$. If $\sigma \in \Gamma'$ leaves the elements of $K'(E)$ fixed, we have $\lambda(\sigma) \in A$, whence $a \in A'$ and $a$ leaves the elements of $E'$ fixed; by Cor. 1 of V, p. 68 we thus have $K'(E) \supset E'$.

We have proved the equality $E' = K'(E)$, whence $\varphi^{-1}(E') = K' \otimes_K E$. If $F$ is a subfield of $N$ containing $K$ and such that $E' = K'(F)$, we have likewise $\varphi^{-1}(E') = K' \otimes_K F$, whence $F = E$.