I am currently a professor at the Universidade Federal de Ciencias da Saude de Porto Alegre, Brazil.
During the year of 2015 I worked under the supervision of Dr. Mark van Hoeij on the problem of finding all subfields of a separable field extension. This visit resulted in a new algorithm with better complexity and run times for computing the subfield lattice.
These results can be found in the paper The Complexity of Computing all Subfields of an Algebraic Number Field, J. Szutkoski and M. van Hoeij, 2016. (Submitted to JSC). An implementation in Magma of this new algorithm can be found here.
Based on the idea of Principal Subfields and fast subfield intersections, we also devised an algorithm for computing all non-equivalent complete decompositions of a univariate rational function f. The algorithm is described in the paper Functional Decomposition using Principal Subfields (to be presented at ISSAC'17) and the implementation can be found here.