Timeline for Why is "h" the notation for class numbers?
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22 events
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| Nov 15, 2023 at 1:56 | comment | added | Jesse Elliott | Having written a book on rings, I always have to explain to my family and friends that they have nothing to do with "rings" in any sense of the English word. And that is true. As much as I love ring theory, I don't like the work "ring" applied to this context. | |
| Jul 1, 2010 at 4:31 | comment | added | Tom Goodwillie | Once, more than half a lifetime ago, I happened to sit next to Andre Weil at a colloquium dinner. Awed and groping for a topic, I asked him why rings are called rings. He offered the very tentative guess that the name came from the idea that you get a ring by making a hole in a field (or should I say a corps or a Koerper). | |
| Mar 6, 2010 at 0:06 | history | edited | Alison Miller |
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| Mar 4, 2010 at 22:57 | vote | accept | KConrad | ||
| Mar 4, 2010 at 21:46 | comment | added | KConrad | See the reference by Cohn that I listed as well in my previous comment. From a websearch I find several places which indicate Hilbert introduced this terminology Zahlring for rings (of alg. integers) in his Zahlbericht. For instance, do a search for W. B. Ewald, "From Kant to Hilbert: a source book in the foundations of mathematics", page 762, footnote a. | |
| Mar 4, 2010 at 21:33 | comment | added | Franz Lemmermeyer | Hilbert did not comment on his choice of words, as far as I know. But Dedekind's symbol O for order looks pretty much a like a ring. I've heard the "cycling back"-explanation very often, but do not know who came up with it first. The word "ring" apparently came later than Hilbert's Zahlring; Fraenkel gave a first set of axioms for rings (differing in content from the modern one) around 1914, I think. Emmy Noether gave rings their modern meaning in the early 1920s. | |
| Mar 4, 2010 at 18:41 | comment | added | Regenbogen | @KConrad. Page 9 of Lemmermayer and Schappacher says that Hilbert used the word Zahlring. But it does not say that he was the first one to use the term ring, and it does not mention your "cycling back in a sense" explanation for an integral dependence relation for an algebraic integer. | |
| Mar 4, 2010 at 18:21 | comment | added | KConrad | See the introduction to the English translation of Hilbert's Zahlbericht. Specifically, look at page 9 on fen.bilkent.edu.tr/~franz/publ/hil.pdf and since Franz is on MO, maybe he can say something more on this. See also Harvey Cohn, "Advanced Number Theory", p. 49. Do a Google search for Zahlring Cohn. | |
| Mar 4, 2010 at 17:59 | comment | added | Regenbogen | @KConrad: A citation for your claim about ring and Zahlring? | |
| Mar 4, 2010 at 15:39 | answer | added | Franz Lemmermeyer | timeline score: 22 | |
| Mar 4, 2010 at 7:47 | answer | added | Olivier | timeline score: 10 | |
| Mar 4, 2010 at 7:22 | answer | added | Mariano Suárez-Álvarez | timeline score: 8 | |
| Mar 4, 2010 at 7:13 | comment | added | Harry Gindi | I had to vote up all of the banter. | |
| Mar 4, 2010 at 7:11 | comment | added | KConrad | I looked in the Disquisitiones (English translation) on Google books and in some sections (though not all) Gauss write equivalence classes of quadratic forms using the letters H, K, and L in various sizes and fonts. See article 259 (pp. 282--283 in the English translation). | |
| Mar 4, 2010 at 7:10 | comment | added | Qiaochu Yuan | Yes, I see; I checked the Disquisitiones after making that comment. For those who are curious the original text is available here: gdz.sub.uni-goettingen.de/dms/load/img | |
| Mar 4, 2010 at 7:06 | comment | added | Mariano Suárez-Álvarez | @Qiaochu: at least, in the §303 of his Disquisitiones he does not use any notation but keeps saying things like 'multitudo classium' ('nombre de classes' in the French edition). I don't think he uses notation elsewhere. | |
| Mar 4, 2010 at 6:59 | comment | added | Mariano Suárez-Álvarez | +1000 ! | |
| Mar 4, 2010 at 6:54 | comment | added | KConrad | The term Zahlring was introduced by Hilbert, and the typos in the previous comment were introduced by me. | |
| Mar 4, 2010 at 6:52 | comment | added | KConrad | Oh, ring comes from the Zahlring, applied to rings like Z[a] when a is an algebraic integer. The idea is that the equations of integral dependence express high powers or an element as integral combinations of a definite set of small powers, thus cycling back in a sense (like a ring). | |
| Mar 4, 2010 at 6:32 | comment | added | Qiaochu Yuan | What notation did Gauss use? | |
| Mar 4, 2010 at 6:31 | comment | added | Mariano Suárez-Álvarez | Good question! Maybe whoever knows the answer knows why a ring is a called a ring, too? | |
| Mar 4, 2010 at 6:27 | history | asked | KConrad | CC BY-SA 2.5 |