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?
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asked Nov 9, 2014 at 20:54
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1$\begingroup$ See also here. $\endgroup$– Andrés E. CaicedoNov 10, 2014 at 7:34
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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
