I read that the primitive element theorem for fields was fundamental in expositions of Galois theory before Emil Artin reformulated the subject. What are the differences between pre and postArtin Galois theory?

The development of Galois theory from Lagrange to Artin by B. Melvin Kiernan, is a history of preArtin Galois theory. 


Two articles by James Pierpont in the first two issues of the annals of math second series give a view of Galois theory as of 1900. They are: Galois' Theory of Algebraic Equations, Ann. of Math. second series, Vol 1 (18991900), 113143, and Galois' Theory of Algebraic Equations. Part II. Irrational Resolvents, Ann. of Math. second series, Vol 2 (19001901), 2256. 


PostArtin, you could read about it in English! No, that's not fair, but few authors writing in English on the "theory of equations" handled it. An exception would be L. E. Dickson, and I looked at one of his books before encountering the socalled modern theory (now aged about 85) of Artin and Emmy Noether, as written up by van der Waerden first. I think I must have read Modern Algebraic Theories by Dickson. Anyway the review of that in http://www.ams.org/journals/bull/19263206/S000299041926043032/S000299041926043032.pdf can give some idea of the good old days, if you can't find the book. By the way, just anecdotal, but G. H. Hardy made some public blunder in Galois theory, so it wasn't really transparent. 

