A formula for the separability idempotent in the tensor square of a field (Q1072596)

A finite dimensional algebra over a field k is said to be separable if there exists an element \(\sum x_ i\otimes y_ i\) of \(A\otimes A\) such that \(\sum ax_ i\otimes y_ i=\sum x_ i\otimes y_ ia\) for all a in A and \(\sum x_ iy_ i=1\). When A is commutative, such an element is unique and an idempotent in \(A\otimes A\). When E is a separable field extension of dimension d over k and a is a primitive element of E over k, the authors give a formula for this separability idempotent in terms of the inverse of the matrix representation of the regular trace form of E over k with respect to the basis \(\{1,a,a^ 2,...,a^{d-1}\}\).