I'm having trouble following the details of the discussion on pages 9 and 10 of Neukirch's algebraic number theory book.
Suppose $L$ is a separable extension of $K$ with degree n. Consider the set of embeddings of $L$ into $\bar K$, the algebraic closure of $K$, that fix $K$ (K-embeddings). Why are there $n$ embeddings in this set?
EDIT: Also, consider some element $x\in L$. Let $d$ be the degree of $L$ over $K(x)$ and $m$ be the degree of $K(x)$ over $K$. Why are the $K$-embeddings of $L$ partitioned by the equivalence relation
$$ \sigma\sim\tau\ \Leftrightarrow\ \sigma x = \tau x $$
into $m$ equivalence classes of $d$ elements each?