×

On a theorem of Scholz on the class number of quadratic fields. (English) Zbl 1062.11071

Let \(K= \mathbb{Q}(\sqrt{pq})\) where \(p\) and \(q\) are distinct primes such that \(p\equiv q\pmod 4\). In the note under review, the author gives another proof of a classical result of A. Scholz, and determines the exact power of 2 dividing the class number of \(K\) using a theorem on the solvability of \(ax^2+ by^2= z^2\).

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ireland, K., and Rosen, M.: A classical introduction to modern number theory. 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York (1990). · Zbl 0712.11001
[2] Lemmermeyer, F.: Reciprocity Laws From Euler to Eisenstein. Springer-Verlag, Berlin (2000). · Zbl 0949.11002
[3] Nemenzo, F.: Quadratic forms, genus groups and rational reciprocity laws. Masteral Thesis, Sophia University (1991).
[4] Nemenzo, F., and Wada, H.: An elementary proof of Gauss’ genus theorem. Proc. Japan Acad., 68A , 94-95 (1992). · Zbl 0763.11042 · doi:10.3792/pjaa.68.94
[5] Scholz, A.; Über die Lösbarkeit der Gleichung \(t^2-Du^2=-4\). Math. Z., 39 , 95-111 (1934).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.