Lemma 10.160.6. Let $p$ be a prime number. Let $k$ be a field of characteristic $p$. There exists a Cohen ring $\Lambda $ with $\Lambda /p\Lambda \cong k$.

**Proof.**
First note that the $p$-adic integers $\mathbf{Z}_ p$ form a Cohen ring for $\mathbf{F}_ p$. Let $k$ be an arbitrary field of characteristic $p$. Let $\mathbf{Z}_ p \to R$ be a flat local ring map such that $\mathfrak m_ R = pR$ and $R/pR = k$, see Lemma 10.159.1. By Lemma 10.97.5 the completion $\Lambda = R^\wedge $ is Noetherian. It is a complete Noetherian local ring with maximal ideal $(p)$ as $\Lambda /p\Lambda = R/pR$ is a field (use Lemma 10.96.3). Since $\mathbf{Z}_ p \to R \to \Lambda $ is flat (by Lemma 10.97.2) we see that $p$ is a nonzerodivisor in $\Lambda $. Hence $\Lambda $ has dimension $\geq 1$ (Lemma 10.60.13) and we conclude that $\Lambda $ is regular of dimension $1$, i.e., a discrete valuation ring by Lemma 10.119.7. We conclude $\Lambda $ is a Cohen ring for $k$.
$\square$

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