Proposition 4

Let p be a prime number, k a field of characteristic p, and let n be an integer $\geq 1$, or $+ \infty$. Then there exists a $p$-ring ($\S$ 2, No. 1, Definition 1) of length n whose residue field is isomorphic to k.

We may regard k as an extension of the residue field $\mathbf{Z}/p\mathbf{Z}$ of the local ring $\mathbf{Z}_{(p)}$. By the corollary to Theorem 1, there exists a local ring B, an inflation of $\mathbf{Z}_{(p)}$, such that $\kappa_{B}$ is isomorphic to k. The local ring $\mathbf{Z}_{(p)}$ is regular and $\{ p \}$ is a system of coordinates of $\mathbf{Z}_{(p)}$. By the corollary to Proposition 2 of No. 2, the ring B is regular and $\{ pl_{B} \}$ is a system of coordinates of B. In other words, B is a discrete valuation ring, with maximal ideal $pB$. The completion C of B is then a $p$-ring of infinite length and the residue field $\kappa_{C}$ is isomorphic to $\kappa_{B}$, hence to k. Moreover, for every integer $n \geq 1$, $C/p^{n}C$ is a $p$-ring of length n, with residue field isomorphic to $\kappa_{C}$, hence to k.

Exercises