Exercise 2, § 2

Let $K, K'$ be two extensions of a field $k$ and $f$ and $f'$ places respectively of $K$ and $K'$ with values in the same algebraically closed field $L$. Suppose that the restrictions of $f$ and $f'$ to $k$ coincide. Show that there exist a composite extension $(F, i, i')$ of $K$ and $K'$ (Algebra, Chapter VIII, § 8) and a place $g$ of $F$ with values in $L$ such that $f(x) = g(i(x))$ for $x \in K$ and $f'(x) = g(i'(x'))$ for $x' \in K'$. (Let $V, V'$ be the rings of $f$ and $f'$, $A$ their common intersection with $k$ and $h : V \otimes_A V' \to L$ the homomorphism such that $h(a \otimes a') = f(a)f'(a')$ for $a \in V$, $a' \in V'$. Using the fact that $V$ and $V'$ are flat $A$-modules ($§ 3$, no. 5, Lemma 1), prove that, if $a \neq 0$, $a' \neq 0$, $a \otimes a' = (a \otimes 1)(1 \otimes a')$ is not a divisor of zero; if $S$ is the multiplicative subset of $B = V \otimes_A V'$ consisting of these elements,

S^{-1}B contains subfields K_1, K'_1 respectively isomorphic to K and K' and is the ring generated by K_1 \cup K'_1. Show that, if q is the kernel of h, there exists a prime ideal p in B such that p \subset q and p \cap S = \varnothing and show that F may be taken as the field of fractions of S^{-1}B/S^{-1}p.)