Lemma 10.50.2. Let K be a field. Let A \subset K be a local subring. Then there exists a valuation ring with fraction field K dominating A.
Proof. We consider the collection of local subrings of K as a partially ordered set using the relation of domination. Suppose that \{ A_ i\} _{i \in I} is a totally ordered collection of local subrings of K. Then B = \bigcup A_ i is a local subring which dominates all of the A_ i. Hence by Zorn's Lemma, it suffices to show that if A \subset K is a local ring whose fraction field is not K, then there exists a local ring B \subset K, B \not= A dominating A.
Pick t \in K which is not in the fraction field of A. If t is transcendental over A, then A[t] \subset K and hence A[t]_{(t, \mathfrak m)} \subset K is a local ring distinct from A dominating A. Suppose t is algebraic over A. Then for some nonzero a \in A the element at is integral over A. In this case the subring A' \subset K generated by A and ta is finite over A. By Lemma 10.36.17 there exists a prime ideal \mathfrak m' \subset A' lying over \mathfrak m. Then A'_{\mathfrak m'} dominates A. If A = A'_{\mathfrak m'}, then t is in the fraction field of A which we assumed not to be the case. Thus A \not= A'_{\mathfrak m'} as desired. \square
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