Timeline for What was Galois theory like before Emil Artin?
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16 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2016 at 13:39 | answer | added | john mangual | timeline score: -1 | |
| Oct 15, 2012 at 0:38 | answer | added | Matt Brin | timeline score: 6 | |
| Jul 2, 2011 at 1:36 | vote | accept | teil | ||
| Jun 1, 2010 at 0:03 | comment | added | KConrad | 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. | |
| May 31, 2010 at 23:54 | comment | added | KConrad | 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. | |
| May 31, 2010 at 20:51 | comment | added | KConrad | 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. | |
| May 31, 2010 at 14:49 | comment | added | Charles Matthews | 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. | |
| May 31, 2010 at 14:22 | comment | added | Charles Matthews | Yes; and see 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. | |
| May 31, 2010 at 13:11 | comment | added | JS Milne | Charles, the independence of multiplicative characters is usually credited to Dedekind. Galois theory is about separable extensions. | |
| May 31, 2010 at 12:42 | history | edited | teil |
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| May 31, 2010 at 12:39 | comment | added | Charles Matthews | 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? | |
| May 31, 2010 at 11:57 | comment | added | JS Milne | 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. | |
| May 31, 2010 at 11:41 | comment | added | Gjergji Zaimi | Maybe you can add the tag ho.history-overview. | |
| May 31, 2010 at 11:24 | answer | added | Gjergji Zaimi | timeline score: 22 | |
| May 31, 2010 at 11:21 | answer | added | Charles Matthews | timeline score: 2 | |
| May 31, 2010 at 10:59 | history | asked | teil | CC BY-SA 2.5 |