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The quadratic fields with discriminant divisible by exactly two primes and with “narrow” class number divisible by 8. (English) Zbl 1071.11060

The author considers quadratic fields \(K\) with discriminants having two prime divisors, and gives an elementary proof of sufficient and necessary conditions for divisibility of the narrow class number of \(K\) by \(8\).

MSC:

11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
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References:

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[2] Basilla, J. M.: Doctoral Thesis. Sophia University. (Preprint).
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[7] Nemenzo, F. R.: On a Theorem of Scholz on the class number of quadratic fields. Proc. Japan Acad., 80A , 9-11 (2004). · Zbl 1062.11071 · doi:10.3792/pjaa.80.9
[8] 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
[9] Rédei, L., and Reichardt, H.: Die Anzahl der durch 4 teilbaren invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers. J. Reine Angew. Math., 170 , 69-74 (1934). · Zbl 0007.39602
[10] Scholz, A.: Über die Lösbarket der Gleichung \(t^2-Du^2=-4\). Math. Z., 39 , 95-111 (1934).
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