Exercise 9, § 3

Let $A$ be a local ring of equal characteristics, $m_A$ its maximal ideal and $k$ a subfield of $A$. Then the residue field $\kappa_A$ is a $k$-algebra. One says that $k$ is a weak field of representatives of $A$ if $\kappa_A$ is formally étale over $k$ (Exercise 8). a) If $A$ possesses a weak field of representatives $k$, then the separated completion $\hat{A}$ of $A$ possesses a unique field of representatives containing the image of $k$ in $\hat{A}$. b) Let $B$ be a local ring contained in $A$, with maximal ideal $m_B = m_A \cap B$. If $\kappa_A$ is a separable extension of the residue field $\kappa_B$ of $B$, every weak field of representatives of $B$ is contained in a weak field of representatives of $A$ (use Exercise 8, c)). c) Let $B$ be a local ring contained in $A$, with maximal ideal $m_B = m_A \cap B$. Suppose moreover that $A$ is of nonzero characteristic $p$ and contains $B^p$. Then there exists a weak field of representatives of $A$ containing a weak field of representatives of $B$.