## Why is “h” the notation for class numbers?

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).

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Good question! Maybe whoever knows the answer knows why a ring is a called a ring, too? – Mariano Suárez-Alvarez Mar 4 2010 at 6:31
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 4 2010 at 6:52
The term Zahlring was introduced by Hilbert, and the typos in the previous comment were introduced by me. – KConrad Mar 4 2010 at 6:54
+1000 ! – Mariano Suárez-Alvarez Mar 4 2010 at 6:59
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 1 2010 at 4:31

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.

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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 4 2010 at 22:56

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)

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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 4 2010 at 17:51
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 4 2010 at 21:26
Are Jacobi's 1837 number theory lectures available somewhere on the web ? – Chandan Singh Dalawat Mar 5 2010 at 3:25
No. But I can send you a pdf file if you want one. The printed version can be ordered at webserver.erwin-rauner.de/tal2007/algor/… (I'm not making any money from this; it's quite difficult to get things like these published at all). – Franz Lemmermeyer Mar 5 2010 at 21:22
Dear Franz, will be extremely grateful for a pdf copy. My email address is <mylastname>@gmail.com. It is great that there are devoted people like you who make these treasures available. – Chandan Singh Dalawat Mar 7 2010 at 6:20
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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.

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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 4 2010 at 8:04