Separable Algebra - Commutative Separable Algebras

Commutative Separable Algebras

If is a field extension, then L is separable as an associative K-algebra if and only if the extension of field is separable. If L/K has primitive element a with irreducible polynomial, then a separability idempotent is given by . The tensorands are dual bases for the trace map: if are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by Tr(x) = . The trace map and its dual bases make explicit L as a Frobenius algebra over K.