Theorem 1

Let A be a separated and complete local ring of characteristic p.

a) Assume $p = 0$ and let $(x_\lambda)_{\lambda \in \Lambda}$ be a family of elements of A whose classes modulo $m_A$ form a transcendence basis of $\kappa_A$ over $\mathbf{Q}$. There exists a unique field of representatives of A containing the elements $x_\lambda$.

b) Assume $p \neq 0$. Let $(x_\lambda)_{\lambda \in \Lambda}$ be a family of elements of A whose classes modulo $m_A$ form a p-basis of $\kappa_A$ (A, V, p. 95). There exists a unique field of representatives of A containing the elements $x_\lambda$. It is a Cohen subring of A.

Assume that one has $p = 0$, so that A is a $\mathbf{Q}$-algebra. Every transcendence basis of $\kappa_A$ over $\mathbf{Q}$ being separating, assertion a) follows from prop. 1 of no. 1 applied to the case $k_0 = \mathbf{Q}$, $K = \kappa_A$.

Assume now that one has $p \neq 0$. Then one has $p1_A = 0$, and every Cohen subring C of A satisfies $pC = 0$. In other words, there is identity between the notions of field of representatives and of Cohen subring of A. Assertion b) then follows from § 2, no. 2, th. 1.