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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: