Congruences modulo 8 for class numbers of general quadratic fields \(\mathbb{Q}(\sqrt{m})\) and \(\mathbb{Q}(\sqrt{-m})\)
The author proves known congruences modulo 8 between class numbers and units of the two fields \(\mathbb{Q}(\sqrt{\pm m})\) [\textit{P.-J. Desnoux}, Manuscr. Math. 62, No. 2, 163--179 (1988; Zbl 0664.12002) and other]. Remarks: (i) Now such congruences are true modulo \(2^{r+2}\) (at least), where \(r\) is the number of odd prime divisors of \(m\). (ii) As many papers are written on the subject, the reviewer takes this opportunity to give less known references: \textit{P.-J. Desnoux} [loc. cit.]; \textit{G. Gras} [Manuscr. Math. 57, 373--415 (1987; Zbl 0599.12016) and Acta Arith. 52, No. 2, 147--162 (1989; Zbl 0618.12004)]; \textit{A. Costa} [Modular forms and class number congruences (Ph. D. Thesis), Univ. Pennsylvania (1989)]; \textit{R. Pioui} [Mesures de Haar \(p\)-adiques et interprétation arithmétique de \(L_2(\chi,s) - L_2(\chi,t)\), \(s,t\in\mathbb{Z}_2\) (Thèse), Univ. Besançon (1990)].
scientific article; zbMATH DE number 4134168