Proposition 5

Let k be a field of characteristic p, and let n be an integer $\geq 1$, or $+ \infty$. There exists a p-ring of length n whose residue field is isomorphic to k.

The ring $W(k)$ of Witt vectors with coefficients in k is an integral separated complete local ring whose residue field is isomorphic to k ($§ 1$, no. 8, prop. 8), and one has $p \cdot 1_{W(k)} \neq 0$ (loc. cit., formula (52)). Let C be a Cohen subring of $W(k)$ (no. 2, th. 1). Then C is a p-ring of length $+ \infty$ whose residue field is isomorphic to k, and, if n is an integer $\geq 1$, the quotient $C/p^nC$ is a p-ring of length n whose residue field is isomorphic to k.