It is well known that Galois, and other mathematicians around that time, considered Galois groups to be permutation groups and approached Galois theory in this manner. At some point the theory took a different angle and it was approached using newly developed field theory. The study of roots of the polynomials was approached with splitting fields etc. This gave a better and more concise statement of the fundamental theory of Galois theory. At which point did this modern approach arise?
A first general treatment (that is, with an abstract notion of field, which is how I understand the question) of Galois theory was given by Heinrich Weber in "Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie" Mathematische Annalen 43 (1893) 548 - 549