It is classical that an algebra, free of finite rank, is etale if and only if its trace form is non-degenerate. This can proved purely by establishing multi-linear algebra properties of the trace form, in particular of its adjunct. No reduction to the field case necessary.
Here are formulas for the separability idempotent:
Furthermore, various different definitions of the different ideal are considered.
The material ows a lot to The Stacks project - 49 Discriminants and Differents.
The text is in preparation since December 2025.