A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective (which based on past teaching I don't entirely share) that the periods worth considering are pre-Artin, classic Artin treatment, and post-Artin. To make the point explicitly, that is to do with the influence of Artin's *Galois Theory* Notre Dame notes, copyright dates 1940 and 1942.

My issues with this periodisation are primarily to do with a wish to have a proper view of innovation, starting with Galois (admitting pre-history evident in Gauss and Abel, solution of the quartic, group theory and other contributions in Lagrange). There is something like this:

*Galois

*Liouville writes up the theory

*French school of group theory and treatment by Camille Jordan

*Riemann surface theory in general, and isogenies of elliptic curves in particular, develop in parallel

*Presumably Hurwitz knew how to connect the dots

*Algebraic number theory uses abelian extensions and Kummer theory extensively

*Hilbert lays conjectural foundations for class field theory, post-Kronecker Jugendtraum and complex multiplication theory, using a version of Galois theory that seems to be much influenced by Hurwitz/Riemann surfaces

*Steinitz, abstract theory of fields, idea of separable extensions clarified

*New expositions from Emmy Noether and Artin in the 1920s (are these documented, though?), against the background of completing proofs of class field theory, and Artin L-functions

*The Inverse Problem for Galois groups is stated and leads to work on invariant theory

*1930s: Galois theory for infinite extensions is enunciated

*C.1940: Tensor products of fields.

This takes us just about to 1940. I think it is a trap to assume Artin in 1940 was lecturing on Galois theory in the precise terms he would have used in the 1920s.

I'd be grateful for help making this tentative timeline more solid. Further interesting things did happen after 1942, but that seems enough for one question.

[Edit:The older question was What was Galois theory like before Emil Artin? - treat my remarks there as tentative.]

**Edit**: Dedekind's contribution should have been on the list. See hss.cmu.edu/philosophy/techreports/184_Dean.pdf about what Dedekind did in his *Vorlesungen*. That article credits Artin with the formulation of the Fundamental Theorem in abstract terms, while crediting Dedekind with the theory for subfields of the complex numbers. In that context it becomes clearer, I think, why "innovation" goes to Dedekind on the lattice-theoretic way of thinking about Galois theory, while Artin might reasonably have thought he was doing exposition (bringing those ideas explicitly into the post-Steinitz era).