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A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\mathcal{O}$ (from Dedekind's use of Ordnung, the German word for order, which was taken from taxonomy in the same way the words class and genus had been stolen for math usage before him), but I was stumped by $h$. Does anyone out there know how $h$ got adopted?

I have a copy of Dirichlet's lecture notes on number theory (the ones Dedekind edited with his famous supplements laying out the theory of ideals), and in there he is using $h$. So this convention goes back at least to Dirichlet -- or maybe Dedekind? -- but is that where the notation starts? And even if so, why the letter $h$?

I had jokingly suggested to the student that $h$ was for Hilbert, but I then told him right away it made no historical sense (Hilbert being too late chronologically).

Alison Miller
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KConrad
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    Good question! Maybe whoever knows the answer knows why a ring is a called a ring, too? – Mariano Suárez-Álvarez Mar 04 '10 at 06:31
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    What notation did Gauss use? – Qiaochu Yuan Mar 04 '10 at 06:32
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    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). – KConrad Mar 04 '10 at 06:52
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    The term Zahlring was introduced by Hilbert, and the typos in the previous comment were introduced by me. – KConrad Mar 04 '10 at 06:54
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    +1000 ! – Mariano Suárez-Álvarez Mar 04 '10 at 06:59
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    @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. – Mariano Suárez-Álvarez Mar 04 '10 at 07:06
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    Yes, I see; I checked the Disquisitiones after making that comment. For those who are curious the original text is available here: http://gdz.sub.uni-goettingen.de/dms/load/img/ – Qiaochu Yuan Mar 04 '10 at 07:10
  • 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). – KConrad Mar 04 '10 at 07:11
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    I had to vote up all of the banter. – Harry Gindi Mar 04 '10 at 07:13
  • @KConrad: A citation for your claim about ring and Zahlring? – Regenbogen Mar 04 '10 at 17:59
  • See the introduction to the English translation of Hilbert's Zahlbericht. Specifically, look at page 9 on

    http://www.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.

     – KConrad Mar 04 '10 at 18:21
  • @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. – Regenbogen Mar 04 '10 at 18:41
  • 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. – Franz Lemmermeyer Mar 04 '10 at 21:33
  • 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.

     – KConrad Mar 04 '10 at 21:46
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    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). – Tom Goodwillie Jul 01 '10 at 04:31
  • 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. – Jesse Elliott Nov 15 '23 at 01:56

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Gauss, in his Disquisitiones, used ad hoc notation for the class number when he needed it. He did not use h. Dirichlet used h for the class number in 1838 when he proved the class number formula for binary quadratic forms. I somewhat doubt that he was thinking of "Hauptform" in this connection - back then, the group structure was not as omnipresent as it is today, and the result that $Q^h$ is the principal form was known (and written additively), but did not play any role. Kummer, 10 years later, used H for the class number of the field of p-th roots of unity, and h for the class number of a subfield generated by Gaussiam periods (and "proved" that $h \mid H$); in the introduction he quotes Dirichlet's work on forms at length.

Franz Lemmermeyer
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    I looked this up. In his 1838 paper on the use of infinite series in number theory (Crelle 18 (1838), 259--274), Dirichlet writes h for the class number. See page 263, where his famous class number formula for imaginary quadratic fields is in the middle of the page. In Crelle vol. 19 page 358 he again introduces h as his notation for the class number.

    There is no L-function notation anywhere. He writes many Dirichlet series and Euler products (as functions of a "continuous positive variable" s), but his notation for what we'd write as L(1,chi) is S.

     – KConrad Mar 04 '10 at 22:56
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I had always thought that it stood for Haupt (principal) because ideals become principal after being raised at the power $h$. However, I don't have any historical reference.

Olivier
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    This is a convincing explanation of what the h should stand for, whether it's actually historically correct or not. I think I'll tell my students this from now on... – Pete L. Clark Mar 04 '10 at 08:04
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F. Cajori gives several pointers in his A history of mathematical notations, Vol. 2, page 40. I think (he's a bit unclear...) he attributes the notation to Kronecker, referring to Dickson's History, Vol. 3, page 93. Dickson, in turn, in page 138 of that volume, tells us that Kronecker uses that notation in [Sitzungsberichte Akad. d. Wissensch. (Berlin, 1885), Vol. II, p. 768-80]

He apparently had introduced numbers $F(d)$, $G(d)$, $E(d)$, and when he needed one more, he used $H$ :P

(Reading on, we find the first appearence of a lowercase $h$ in Dickson referring to a paper of Weber (Göttingen Nachr., 1893, 138--147, 263--4), so---since Dickson uses notation from the papers he is quoting, we can blame Weber for the change of case)

Mariano Suárez-Álvarez
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    These citations to Kronecker and Weber involve things written in the 1880s and 1890s, but as I wrote in my original question the use of h for class numbers goes back at least to Dirichlet's lectures on number theory, which definitely preceded that. In Section 95 (p. 242) of Dirichlet--Dedekind's Zahlentheorie, Dirichlet is writing h for class numbers and in section 97 he determines the class number of Q(i) by writing "h = (4/pi)(1 - 1/3 + 1/5 - 1/7 + ...) = 1". – KConrad Mar 04 '10 at 17:51
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    Dirichlet's lectures were heavily edited by Dedekind, hence cannot be used as a reliable source of what goes back to Dirichlet and what does not. But Dirichlet used h in his articles, whereas Gauss did not. Neither did Jacobi in his number theory lectures in 1837. So unless the h can be found in Legendre (which I don't believe; he had the "wrong" notion of class number anyway), the credit for introducing h goes to Dirichlet. – Franz Lemmermeyer Mar 04 '10 at 21:26
  • Are Jacobi's 1837 number theory lectures available somewhere on the web ? – Chandan Singh Dalawat Mar 05 '10 at 03:25
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    No. But I can send you a pdf file if you want one. The printed version can be ordered at http://webserver.erwin-rauner.de/tal2007/algor/ign_publ.htm#H62 (I'm not making any money from this; it's quite difficult to get things like these published at all). – Franz Lemmermeyer Mar 05 '10 at 21:22
  • Dear Franz, will be extremely grateful for a pdf copy. My email address is @gmail.com. It is great that there are devoted people like you who make these treasures available. – Chandan Singh Dalawat Mar 07 '10 at 06:20
  • Franz, if the author is open to all, I, too, would love to have a PDF copy. My e-mail address is .@tcu.edu. – LSpice Mar 30 '10 at 23:20