Lemma 10.119.13. Let R be a Noetherian local domain with fraction field K. Assume that R is not a field. Let L/K be a finitely generated field extension. Then there exists discrete valuation ring A with fraction field L which dominates R.
Proof. If L is not finite over K choose a transcendence basis x_1, \ldots , x_ r of L over K and replace R by R[x_1, \ldots , x_ r] localized at the maximal ideal generated by \mathfrak m_ R and x_1, \ldots , x_ r. Thus we may assume K \subset L finite.
By Lemma 10.119.1 we may assume \dim (R) = 1.
Let A \subset L be the integral closure of R in L. By Lemma 10.119.12 this is Noetherian. By Lemma 10.36.17 there is a prime ideal \mathfrak q \subset A lying over the maximal ideal of R. By Lemma 10.119.7 the ring A_{\mathfrak q} is a discrete valuation ring dominating R as desired. \square
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