I am reading a set of Galois theory notes, and I don't understand the proof of the following theorem:
Let $f(X)$ be in $K[X]$, let $L/K$ be a splitting field for $f(X)$. Let $s\colon K\to M$ be any embedding such that $sf$ splits into linear factors in $M[X]$. Then the following hold:
- There exists at least one embedding $t\colon L\to M$ extending $s$.
- The number of embeddings $t$ as in (1) is at most $[L:K]$, with equality holding if $f(X)$ has no repeated roots in $L$, i.e., if it splits into distinct linear factors.
For now I am only asking for a proof of the first part of (2). The more intuitive it is, the better.