Coefficient fields need not be unique

By [L1], $A$ is local with maximal ideal $(t)$ and residue field $A\mathrm{/}(t)\cong k(u)$. The standard inclusion of $k(u)$ into $A$ is therefore a coefficient field, in line with [L2].

Consider the subfield $K^{\mathrm{\prime }}=k(u+t)\subseteq A$. For every nonzero polynomial $q(Z)\in k[Z]$, the residue of $q(u+t)$ modulo $t$ is $q(u)$, which is nonzero in $k(u)$. Hence $q(u+t)$ is a unit of $A$, so every rational function in $u+t$ lies in $A$ and $K^{\mathrm{\prime }}$ is indeed a subfield. Its residue image is again $k(u)$ because $u+t\mapsto u$.

The two coefficient fields are distinct: if $u+t$ lay in the constant field $k(u)$, then subtracting $u$ would place $t$ in $k(u)$, but every nonzero element of $k(u)$ is a unit in $A$ whereas $t$ lies in the maximal ideal. Thus $k(u+t)\mathrm{\ne }k(u)$.

Therefore coefficient fields in a complete equicharacteristic local ring need not be canonical.