Proposition 8

Let A be a field of characteristic p. The ring $W(A)$ is a separated and complete integral local ring, with maximal ideal $V_1(A)$ and residue field isomorphic to A. If the field A is perfect, the ring $W(A)$ is a discrete valuation ring, and its maximal ideal is $p.W(A)$.

The homomorphism $\Phi_0$ defines an isomorphism of $W(A)/V_1(A)$ onto $A$ (no 7, example 1). The ideal $V_1(A)$ of $W(A)$ is therefore maximal. Since the ring $W(A)$ is separated and complete for the $V_1(A)$-adic topology (prop. 6, $b$), it is a local ring, with maximal ideal $V_1(A)$ (III, § 2, no 13, prop. 19).

Let $a$ and $b$ be two nonzero elements of $W(A)$. There exist integers $m \geqslant 0$ and $n \geqslant 0$, and elements $a' = (a'_n)_{n \in \mathbf{N}}$ and $b' = (b'_n)_{n \in \mathbf{N}}$ of $W(A)$ such that $a = V^m(a')$, $b = V^n(b')$ and that the elements $a'_0$ and $b'_0$ of $A$ are nonzero. Then the component of index $m + n$ of $a \times b$ is equal to the component of index 0 of $F^n(a') \times F^m(b')$ (formula (53)), that is to say to ${a'_0}^{p^n} {b'_0}^{p^m}$ (formula (51) and No. 3, Example 2). Consequently $a \times b$ is nonzero and $W(A)$ is an integral domain.

If the field $A$ is perfect, the maximal ideal $V_1(A)$ of $W(A)$ is equal to $p.W(A)$ (Prop. 7, b)) and consequently W(A) is a discrete valuation ring (VI, § 3, No. 6, Prop. 9, c)).