Jacobson-Bourbaki theory with parameters (Q1288754)

Given a finite field extension \(L/K\), the classical Jacobson-Bourbaki correspondence is that between the lattice of intermediate fields \(M\), \(K\subseteq M \subseteq L\), and the lattice of the algebras of linear endomorphisms of \(L\) over \(M\). This correspondence can be used to derive standard Galois theory as a particular case. The purpose of the paper under review is to generalize the Jacobson-Bourbaki correspondence to one between the lattice of \(K\)-tensor products \(V\otimes _K M\) of a fixed \(K\)-vector space \(V\) with all intermediate fields \(K\subseteq M\subseteq L\), and the lattice of the algebras of linear endomorphisms of \(V\otimes _K L\) over those intermediate fields \(M\), which are simultaneously linear endomorphisms of \(V\otimes _K L\) over the ring \(\hom _K (V, V)\). This correspondence is an inclusion reversing bijection. In the last section, the author provides some examples illustrating the correspondence obtained.