Timeline for Dimension of central simple algebra over a global field "built using class field theory".
Current License: CC BY-SA 4.0
11 events
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| Feb 10, 2022 at 0:23 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
fixed broken link
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| Jun 18, 2010 at 16:19 | comment | added | Victor Protsak | @Pete: In fairness to Springer, they've digitized a gazillion books recently, including all LNM and all books published or reprinted since 2006 (available electronically in their entirety to subscribers). I think they just haven't gotten around to some of the older titles. The one I miss is Jantzen's "Einhuellende Algebren halfeinfacher Lie-Algebren". | |
| Jun 17, 2010 at 17:23 | comment | added | Pete L. Clark | @Boyarsky: Thanks so much for the link! I bought a slew of books recently and Pierce was on my list...but it is out of print. (WTF: a fantastic book in the GTM series? My respect for the publishing companies has dipped even lower.) | |
| Jun 15, 2010 at 15:40 | comment | added | Kevin Buzzard | PS @all: Pierce would be where I would have looked had asking here not worked so efficiently. Problem is that Pierce isn't on my office shelf :-( | |
| Jun 15, 2010 at 15:40 | comment | added | Boyarsky | @Pete: I stand corrected: the Chinese webpage ishare.iask.sina.com.cn/f/6403052.html provides a free copy of Pierce's book as a .djvu file (click on the downwards-pointing green arrow there, not the upwords one), so the book exists after all. The Theorem in section 18.6 is the desired result (stated only for number fields solely because Pierce only reviews the basics of number theory in the char. 0 case; the method of proof via Grunwald-Wang is the same as in the general case). | |
| Jun 15, 2010 at 15:39 | comment | added | Kevin Buzzard | @D. Savitt---apologies. My above comments are wrong and your reference is indeed fine (modulo the fact that there's no reference for the proof given). | |
| Jun 15, 2010 at 15:25 | comment | added | Boyarsky | @Pete: The book of Gille-Szamuely punts to elsewhere for getting the cyclic splitting field of correct degree: see Remarks 6.5.5 and 6.5.6. Amusingly, for global function fields they refer to Weil's "Basic Number Theory" without saying where in that dense tome it is to be found, and my recollection is that Weil's book does not handle the degree of the cyclic splitting field. But they also refer the reader to Pierce's "Associative Algebras" for proofs for global fields (or at least number fields). But that isn't freely available on the Internet, and so by modern standards it does not exist. | |
| Jun 15, 2010 at 15:08 | comment | added | Pete L. Clark | @D: Yes, that's right. But Boyarsky is also right: I don't give any reference to a proof, not even a [?]. In my defense, this is because for me, Brauer groups are sort of a prelude to what I really want to talk about. (I do think that a standard reference on CSA's should give a proof: for instance, I would be surprised if the result cannot be found in Pierce or Gille-Szamuely.) | |
| Jun 15, 2010 at 14:59 | comment | added | D. Savitt | Isn't Pete's property Br(1) for a field K the property that period=index holds for all finite extensions of K? | |
| Jun 15, 2010 at 14:04 | comment | added | Kevin Buzzard | @D. Savitt: that my question is the period-index question for local and global fields does indeed seem to be the case, although I don't understand why the definition of I(eta) and M(eta) in Pete's example 1.1.2 coincide with his earlier definitions. But Pete is only claiming that period=index for F the rationals, and I asked about the case of a general global field. It's comforting to see it in print for these cases though ;-) because of course the moment you realise you don't know a proof and the standard references you pick up don't give one either, you begin to think it might be wrong... | |
| Jun 15, 2010 at 11:17 | history | answered | D. Savitt | CC BY-SA 2.5 |