Generating subfields

Given a field extension K/k of degree n we are interested in finding the subfields of K containing k. There can be more than polynomially many subfields. We introduce the notion of generating subfields, a set of up to n subfields whose intersections give the rest. We provide an efficient algorithm which uses linear algebra in k or lattice reduction along with factorization in any extension of K. Implementations show that previously difficult cases can now be handled.