A note on Hasse's theorem concerning the class number formula of real quadratic fields
Let p be a prime with \(p\equiv 1 (mod 4)\) and h be the class number of the real quadratic field \({\mathbb{Q}}(\sqrt{p})\). Let \(\epsilon>1\) be a fundamental unit of \({\mathbb{Q}}(\sqrt{p})\). As is well-known, the Dirichlet class number formula is given in the form \[ \epsilon^ h=\prod_{b}\sin \pi b/p\quad /\prod_{a}\sin \pi a/p\quad, \] where a and b runs over the quadratic residues and quadratic non-residues between 0 and p/2, respectively. \(\epsilon^ h\) is written in the form \(u+v\sqrt{p},\quad u,v\in {\mathbb{Q}}.\) The explicit formula of u and v was given by H. Hasse. In this paper the author proves an alternative form of Hasse's theorem, which is slightly simpler in structure than Hasse's one.
class number of the real quadratic field
scientific article; zbMATH DE number 3853204