Lemma 15.38.4. Let $k$ be a field. Let $(A, \mathfrak m, \kappa )$ be a complete local $k$-algebra. If $\kappa /k$ is separable and $A$ regular, then there exists an isomorphism of $A \cong \kappa [[t_1, \ldots , t_ d]]$ as $k$-algebras.

Proof. Choose $\kappa \to A$ as in Lemma 15.38.3 and apply Algebra, Lemma 10.160.10. $\square$

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