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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?

teil
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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
  • Is that right? I believe that the independence of multiplicative characters was an innovation. It would have replaced an explicit calculation of determinants, which would be group determinants, which would have been known about in principle since Frobenius ... Anyway that locates the part of the proof of the fundamental theorem where something had to happen (Kaplansky showed that a relatively small amount of something serious proves your adjunction a duality). Speaking of primitive elements, the tacit assumption that extensions are separable would have been a feature before Steinitz? – Charles Matthews May 31 '10 at 12:39
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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
  • Yes; and see http://www.math.uconn.edu/~kconrad/math316/linearchar.pdf for the attribution of the use of the independence in the proof to Artin (with criticism, too). I'm reminded that the Oxford course on Galois theory was or is 16 lectures, the Cambridge one was or is 24 lectures. One clear difference was or is whether separability is treated seriously, so that for example one can give an example of a finite extension without a primitive element. – Charles Matthews May 31 '10 at 14:22
  • Well, I'm now confused as to whether what Dedekind proved was morally the evaluation of the "Dedekind determinant", or not, so I'd better adjourn my commentary. – Charles Matthews May 31 '10 at 14:49
  • Charles, I looked at my .pdf file and you misunderstood what I wrote. I say (at the end of the introduction) that Artin used linear independence of characters in his treatment, but I didn't mean to suggest that lin. indep. was due to him. The study of characters on general finite abelian groups goes back, I believe, to Weber. (Of course there were concrete antecedents, such as Dirichlet characters and the Legendre symbol.) The linear independence of characters is so closely related to Dedekind's group determinant that the lin. indept. is surely due to Dedekind or someone before him. – KConrad May 31 '10 at 20:51
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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 01 '10 at 00:03

4 Answers4

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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.

Glorfindel
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Gjergji Zaimi
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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.

Assad Ebrahim
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Matt Brin
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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 Bulletin of the AMS (Bull. Amer. Math. Soc. 32 (1926), 707-710) 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.

David Roberts
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Charles Matthews
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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 http://www.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 03 '10 at 09:50
  • Does not van der Waerden attribute the treatment to Artin? – Igor Rivin Dec 03 '16 at 18:35
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it was called "theory of equations" in certain places in English. one learned how to solve cubic and quartic and other special types. Galois theory was taught literally as the study of symmetric polynomials of the roots - the invariants and coinvariants

here's one

Galois' own writings are available online. His total writings total less than 100 pages (only some of it on theory of permutation "groups" for solving equations) and changed the course of mathematics.

john mangual
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