The Stacks project

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 $\overline{F}$ having the following property:

  1. $\beta _1 \in \overline{F}$ is a root of $P_1$; let $\varphi _1 : K_1 \to \overline{F}$ be the homomorphism over $F$ sending $\alpha _1$ to $\beta _1$,

  2. $\beta _2 \in \overline{F}$ is a root of $P_2^{\varphi _1}$; let $\varphi _2 : K_2 \to \overline{F}$ be the homomorphism extending $\varphi _1$ sending $\alpha _2$ to $\beta _2$,

  3. and so on until,

  4. $\beta _ n \in \overline{F}$ is a root of $P_ n^{\varphi _{n - 1}}$.

In each step the homorphism $\varphi _ i$ exists and is unique because $K_ i = K_{i - 1}[x]/(P_ i)$ and $\beta _ i$ is a root of $P_ i^{\varphi _{i - 1}}$.

Proof. The map from left to right is discussed above the lemma. Let $(\beta _1, \ldots , \beta _ n)$ be an element of the right hand side. Then we let $\varphi : K = K_ n \to \overline{F}$ be the unique homorphism extending $\varphi _{n - 1}$ sending $\alpha _ n$ to $\beta _ n$. Uniqueness implies that the two constructions are mutually inverse. $\square$

Comments (2)

Comment #11057 by david on

This doesn't make sense to me. What is the right hand side, exactly? It seems to depend on , which is an element of the left hand side. Thus, the "bijection" is between a set and a set that depends on the chosen element of

There are also:

  • 8 comment(s) on Section 9.12: Separable algebraic extensions

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