Theorem 1

Let A be a local, separated and complete ring, whose residue field k is of characteristic p. Let $\pi$ be the canonical mapping of A onto k, and let S be a subset of A, such that $\pi$ induces a bijection of S onto a $p$-basis of k (A, V, p. 95). a) There exists one and only one Cohen subring C of A containing S. b) The subring C of A is closed, and the $pC$-adic topology of C is induced by the $m_A$-adic topology of A. c) Every closed subring $A'$ of A, containing S, and such that $A = A' + m_A$, contains C.

A) Particular case: $m_A$ nilpotent

Let n be a positive integer such that $m_A^{n+1} = \{0\}$. If $\Phi_n$ is the nth Witt polynomial ($§ 1$, No. 1), the mapping $u : [a_0, ..., a_n] \mapsto \Phi_n(a_0, ..., a_n)$ is a ring homomorphism of $W_{n+1}(A)$ into A ($§ 1$, No. 7). Let $B_n$ be the image of $u$ and let $C_n$ be the subring of A generated by $B_n \cup S$.