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).



Why is "h" the notation for class numbers?
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.


