The Complexity of Computing all Subfields of an Algebraic Number Field
  For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity. 
Jonas Szutkoski and Mark van Hoeij
cs.SC math.NT