Counting elements with given trace and norm in étale algebras

In this note, we estimate the number of elements with given trace and norm in a finite étale algebra over a finite field. Our bound extends previous results from finite extension fields to arbitrary finite étale algebras. The idea is to reduce the general case to the split case, where the problem becomes counting rational points on a toric Calabi–Yau hypersurface and where there are sharp estimates available.