Proposition 4

Let $K$ be a field, $f$ a place of $K$ with values in a field $L$ and $K'$ an extension of $K$. Then there exists an extension $L'$ of $L$ and a place $f'$ of $K'$ with values in $L'$ which extends $f$. If $x_1, \ldots, x_n$ are elements of $K$ which are algebraically independent over $K$ and $a_1, \ldots, a_n$ any elements $\in L$, $f'$ may be chosen such that $f(x_i) = a_i$ for $1 \leq i \leq n.$

Let $V$ be the ring off, $g$ the restriction off to $V$ and $g'$ the extension of $g$ to $V[x_1, \ldots, x_n]$ such that $g'(x_i) = a,$ for $1 \leq i \leq n.$ It is sufficient to take $L'$ to be an algebraic closure of $L$ and apply Proposition 3 to $g'$ and $L'$: we obtain a placef': $K' \to \tilde{L}'$ which extends $g'$; if $x \in K - V,$ then $x^{-1} \in m(V),$ whence $f(x^{-1}) = g(x^{-1}) = 0$ and $f'(x) = \infty = f(x);$ hencef' extendsf.