Remark

Let $K$ and $K'$ be two fields and $u$ a homomorphism of $K$ into $K'$. Let $K_s$ (resp. $K'_s$) be a separable closure (V, p. 45, Prop. 14) of $K$ (resp. $K'$) and $\Pi$ (resp. $\Pi'$) the Galois group of $K_s$ over $K$ (resp. $K'_s$ over $K'$). Since $K_s$ is a separable algebraic extension of $K$ and the extension $(K'_s, u)$ of $K$ is separably closed, there exists (V, p. 45, Cor.) a homomorphism $v$ of $K_s$ into $K'_s$ extending $u$. From $v$ we obtain a continuous homomorphism $v^*$ of $\Pi'$ into $\Pi$. Let $v_1$ be another extension of $u$; since $K_s$ is a quasi-Galois extension of $K$, there exists an element $a_s$ of $\Pi$ such that $v_1 = v \circ a_s$. We conclude that $v_1^*(\tau) = \sigma_0^{-1} v^*(\tau) \sigma_0$ for all $\tau \in \Pi$.