Let $A$ be a local ring. Say a monic $f\in A[x]$ is unramifiable if it admits a simple root in its universal splitting algebra (equivalently, it admits a simple root in any ring over which it splits). Say a local ring is separably closed if every unramifiable polynomial admits a simple root. By a result of Wraith, the separably closed local rings are precisely the strictly Henselian ones.

A separably closed field is precisely one without separable extensions.

1. Is it also true that (injective?) separable algebras over a separably closed local ring are isomorphisms?
2. Is there an elementary characterization of (injective?) separable algebras over a local ring, perhaps as those where every element upstairs is a simple root of an unramifiable polynomial downstairs?

Let $A$ be a local ring. Say a monic $f\in A[x]$ is unramifiable if it admits a simple root in its universal splitting algebra (equivalently, it admits a simple root in any ring over which it splits). Say a local ring is separably closed if every unramifiable polynomial admits a simple root. By a result of Wraith, the separably closed local rings are precisely the strictly Henselian ones.

A separably closed field is precisely one without separable extensions.

Is it also true that (injective?) separable algebras over a separably closed local ring are isomorphisms?

Let $A$ be a local ring. Say $f\in A[x]$ is unramifiable if it admits a simple root in its universal splitting algebra (equivalently, it admits a simple root in any ring over which it splits). Say a local ring is separably closed if every unramifiable polynomial admits a simple root. By a result of Wraith, the separably closed local rings are precisely the strictly Henselian ones.

A separably closed field is precisely one without separable extensions.

1. Is it also true that (injective?) separable algebras over a separably closed local ring are isomorphisms?
2. Is there an elementary characterization of (injective?) separable algebras over a local ring, perhaps as those where every element upstairs is a simple root of an unramifiable polynomial downstairs?

Let $A$ be a local ring. Say a monic $f\in A[x]$ is unramifiable if it admits a simple root in its universal splitting algebra (equivalently, it admits a simple root in any ring over which it splits). Say a local ring is separably closed if every unramifiable polynomial admits a simple root. By a result of Wraith, the separably closed local rings are precisely the strictly Henselian ones.

A separably closed field is precisely one without separable extensions.

1. Is it also true that (injective?) separable algebras over a separably closed local ring are isomorphisms?
2. Is there an elementary characterization of (injective?) separable algebras over a local ring, perhaps as those where every element upstairs is a simple root of an unramifiable polynomial downstairs?