Lemma 15.113.2. Let $A$ be a discrete valuation ring with field of fractions $K$. Let $A^\wedge$ be the completion of $A$ with fraction field $K^\wedge$. If $M/K^\wedge$ is a finite separable extension, then there exists a finite separable extension $L/K$ such that $M = K^\wedge \otimes _ K L$.

Proof. Note that $A^\wedge$ is a discrete valuation ring too (by Lemmas 15.43.4 and 15.43.1). In particular $A^\wedge$ is a domain. The proof will work more generally for Noetherian local rings $A$ such that $A^\wedge$ is a local domain of dimension $1$.

Let $\theta \in M$ be an element that generates $M$ over $K^\wedge$. (Theorem of the primitive element.) Let $P(t) \in K^\wedge [t]$ be the minimal polynomial of $\theta$ over $K^\wedge$. Let $\pi \in \mathfrak m_ A$ be a nonzero element. After replacing $\theta$ by $\pi ^ n\theta$ we may assume that the coefficients of $P(t)$ are in $A^\wedge$. Let $B = A^\wedge [\theta ] = A^\wedge [t]/(P(t))$. Note that $B$ is a complete local domain of dimension $1$ because it is finite over $A$ and contained in $M$. Since $M$ is separable over $K$ the element $\theta$ is not a root of the derivative of $P$. For any integer $n$ we can find a monic polynomial $P_1 \in A[t]$ such that $P - P_1$ has coefficients in $\pi ^ nA^\wedge [t]$. By Krasner's lemma (Lemma 15.113.1) we see that $P_1$ has a root $\beta$ in $B$ for $n$ sufficiently large. Moreover, we may assume (if $n$ is chosen large enough) that $\theta - \beta \in \pi B$. Consider the map $\Phi : A^\wedge [t]/(P_1) \to B$ of $A^\wedge$-algebras which maps $t$ to $\beta$. Since $B = \pi B + \sum _{i < \deg (P)} A^\wedge \theta ^ i$, the map $\Phi$ is surjective by Nakayama's lemma. As $\deg (P_1) = \deg (P)$ it follows that $\Phi$ is an isomorphism. We conclude that the ring extension $L = K[t]/(P_1(t))$ satisfies $K^\wedge \otimes _ K L \cong M$. This implies that $L$ is a field and the proof is complete. $\square$

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