Let E be an elliptic curve defined over the rational field Q. We consider
the rank of E(K) as K ranges over certain families of cubic extensions
Given a quadratic field F=Q(sqrt(D)), the families we consider are all
fields K whose Galois closures contain F. We give various criteria which
for such a family to contain an infinitely many fields for which the
rank of E(K) is greater
than the rank of E(Q). The cases D=1 and D=-3 are of special interest.