Does anyone know how to fill the details of Lemma 2.3.4 from the book "Field Arithmetic" by M. D. Fried and M. Jarden. The statement of the lemma goes as follows:
Let $v$ be a discrete valuation of a field $E$, $h\in O_v[X]$ a monic irreducible polynomial of degree $n$, $x$ a root of $h(X)\in\tilde{E}(x)$. Suppose the residue polynomial $\bar h(X)$ is separable. Then author claims that $v$ is unramified in $F$.
A part of the proof says that the residue map $O_v\to \bar E_v$ can be extended to a place $\varphi_i$ of $F$ with $\varphi_i(x)=a_i$. Here for each $i$ between $1$ and $r$ $a_i$ is a root of $h_i(X)$ in $(\bar E_v)_s$.
How can one show the claim above? The book says it follows from the previous result that I wrote in A homomorphism that extends to a place or an embedding but what is the reasoning that it do not extends to a embedding?
