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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 post-Artin Galois theory?

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Maybe you can add the tag ho.history-overview. –  Gjergji Zaimi May 31 '10 at 11:41
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Actually, it wasn't all that different, except that you first proved the primitive element theorem, and then proved things by choosing a primitive element. Artin disliked having to make a choice, and his main contribution was show that you can do Galois theory without choosing a primitive element. It's not obvious to me that this makes things easier or better. You can find the old approach in A.A. Albert's book on algebra. –  JS Milne May 31 '10 at 11:57
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Charles, the independence of multiplicative characters is usually credited to Dedekind. Galois theory is about separable extensions. –  JS Milne May 31 '10 at 13:11
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I checked some references and Milne is right: Dedekind is the person who introduced characters on general finite abelian groups. Weber simply popularized them further in his own books. –  KConrad May 31 '10 at 23:54
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I would suggest, based only on gut feeling here, that Artin's innovation insofar as characters are concerned was seeing that letting characters take values in any field, not just (as with Dedekind) the complex numbers, would be useful to Galois theory. I'm not sure that Dedekind could conceive such a thing, as the only fields in his day where subfields of C, fields of functions (as on a curve), and finite fields. In Dedekind's time, number fields were subfields of C. General fields were introduced by Steinitz in the early 20th century. –  KConrad Jun 1 '10 at 0:03

3 Answers 3

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

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I've now seen some negative comments about a few points in Kiernan. –  Charles Matthews Jun 3 '10 at 9:52

Post-Artin, 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 so-called 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/1926-32-06/S0002-9904-1926-04303-2/S0002-9904-1926-04303-2.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.

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There is also Dickson's first book (according to Wikipedia) "Linear Groups with an Exposition of Galois Field Theory", first published in 1901. Ancient! Oh whoops. Actually, the book doesn't seem to feature any Galois theory, but rather "Galois Field"="Finite Field". It's a book about linear groups over finite fields. –  Matthew Morrow May 31 '10 at 11:35
    
Remarks: Van der Waerden's treatment in early editions of his Modern Algebra doesn't fit the pattern: uses primitive elements (perhaps to be constructive, as he notes in his intro that he wants to be). The Dedekind determinant issue does appear to be close enough to linear independence of 1-D characters of G in K*: abelianise G, wlog, and then if characters are dependent the group determinant can't have the basis of eigenvectors we know. Clear to Artin, doubtless. –  Charles Matthews May 31 '10 at 15:56
    
But it appears that the linear independence was made more nearly explicit by Dedekind (see p.7 of hss.cmu.edu/philosophy/techreports/184_Dean.pdf) in his Vorlesungen. That article also credits Artin with the formulation of the Fundamental Theorem in abstract terms, while crediting Dedekind with the theory for subfields of the complex numbers. –  Charles Matthews Jun 3 '10 at 9:50

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 (1899-1900), 113-143,

and

Galois' Theory of Algebraic Equations. Part II. Irrational Resolvents, Ann. of Math. second series, Vol 2 (1900-1901), 22-56.

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