Exercise 1, § 2

Let $\Omega$ be an extension of a field $K$, $E$ and $F$ two extensions of $K$ contained in $\Omega$ and linearly disjoint over $K$ and $L$ an extension of $K$ contained in $E$. Let $f$ be a place of $E$ with values in $L$ which reduces to the identity automorphism on $K$. Show that there exists a unique place of $K(E \cup F)$ with values in $K(L \cup F)$, extending $f$ and reducing to the identity automorphism on $F$. (If $A$ is the ring of $f$, note that $f$ can be extended uniquely to a homomorphism $g$ from $F[A]$ to $K(L \cup F)$ reducing to the identity on $F$ and that, if $m$ is the ideal off, every element $z \in F[A]$ such that $g(z) = 0$ can be written as $xu$, where $x \in m$, $u \in F[A]$ and $g(u) \neq 0$. Conclude by noting that $K(E \cup F)$ is the field of fractions of $F[A]$.)