Lemma 9.12.8. In Situation 9.12.7 the correspondence

$\mathop{\mathrm{Mor}}\nolimits _ F(K, \overline{F}) \longrightarrow \{ (\beta _1, \ldots , \beta _ n)\text{ as below}\} , \quad \varphi \longmapsto (\varphi (\alpha _1), \ldots , \varphi (\alpha _ n))$

is a bijection. Here the right hand side is the set of $n$-tuples $(\beta _1, \ldots , \beta _ n)$ of elements of $\overline{F}$ such that $\beta _ i$ is a root of $P_ i^\varphi$.

Proof. Let $(\beta _1, \ldots , \beta _ n)$ be an element of the right hand side. We construct a map of fields corresponding to it by induction. Namely, we set $\varphi _0 : K_0 \to \overline{F}$ equal to the given map $K_0 = F \subset \overline{F}$. Having constructed $\varphi _{i - 1} : K_{i - 1} \to \overline{F}$ we observe that $K_ i = K_{i - 1}[x]/(P_ i)$. Hence we can set $\varphi _ i$ equal to the unique map $K_ i \to \overline{F}$ inducing $\varphi _{i - 1}$ on $K_{i - 1}$ and mapping $x$ to $\beta _ i$. This works precisely as $\beta _ i$ is a root of $P_ i^\varphi$. Uniqueness implies that the two constructions are mutually inverse. $\square$

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