Restriction partitions embeddings in a finite tower into extension fibres

Restricting an $F$-embedding $L\to \mathrm{\Omega }$ to $K$ gives an $F$-embedding by [L1].

To extend a chosen $\sigma$, take a finite $K$-basis $a_1,\dots ,a_r$ of $L$ by [L2] and put $K_i=K(a_1,\dots ,a_i)$. Starting with ${\tau }_0=\sigma$, regard each ${\tau }_{i-1}$ as an isomorphism onto its image, transport the minimal polynomial of $a_i$ over $K_{i-1}$ along it by [L3], choose a root in $\mathrm{\Omega }$ by [L5], and extend ${\tau }_{i-1}$ to $K_i$ by [L4]. After finitely many steps, $K_r=L$, so every $\sigma$ has an extension and the restriction map is surjective.

Identify $K$ with $\sigma (K)$. The extensions of $\sigma$ are exactly the $\sigma (K)$-embeddings of the scalar-transported copy of $L$ into $\mathrm{\Omega }$. Since $\mathrm{\Omega }$ is algebraically closed and algebraic over $\sigma (K)$, it is an algebraic closure of that copy of $K$.

Transporting scalars and maps along the isomorphism $K\to \sigma (K)$ identifies the embeddings in step 1.3 with embeddings of $L\mathrm{/}K$ into an algebraic closure of $K$. By [L6], their number is the closure-independent value $[L:K{]}_s$. Thus every fibre of restriction has that cardinality.

In particular, transport along an isomorphism between two embedded copies of $K$ gives a bijection between their restriction fibres.