Algebraic extensions of subgroups in free groups mimic the corresponding constructions in field theory. Finite automata (Stalling's foldings) provide an adequate tool to study such algebraic extensions. I will show how one can apply some ideas and techniques from the classical field theory in studying subgroups of free groups. In particular, I am going to discuss some new results on Hanna Neumann conjecture, test elements, and commensurators of subgroups.
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.