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.