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On the Divisibility of the Class Numbers of Q(√−p) and Q(√−2p) by 16.

Published online by Cambridge University Press:  20 November 2018

Philip A. Leonard  and
Kenneth S. Williams
Philip A. Leonard
Affiliation:
Arizona State University, Tempe, Arizona, U.S.A. 82581
Kenneth S. Williams
Affiliation:
Carleton University Ottawa, Ontario, CanadaK1s 5b6
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Abstract

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Let h(m) denote the class number of the quadratic field Q(√m). In this paper necessary and sufficient conditions for h (m) to be divisible by 16 are determined when m = −p, where p is a prime congruent to 1 modulo 8, and when m = −2p, where p is a prime congruent to ±1 modulo 8.

Keywords

12A2512A50Key wordsand phrasesclass numberimaginary quadratic fieldbinary quadratic forms
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 2 , 01 June 1982 , pp. 200 - 206
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Bauer, Helmut, Zur Berechnung der 2-Klassenzahl der quadratische Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilem, J. Reine Angew. Math. 248 (1971), 42-46.Google Scholar
2. Berndt, Bruce C., Classical theorems on quadratic residues, L'Enseignement Math. 22 (1976), 261-304.Google Scholar
3. Brown, Ezra, The class number of Q(√-p), for p ≡ 1 (mod 8) a prime, Proc. Amer. Math. Soc. 31 (1972), 381-383.Google Scholar
4. Brown, Ezra, The power of 2 dividing the class-number of a binary quadratic discriminant, J. Number Theory 5 (1973), 413-419.Google Scholar
5. Brown, Ezra, Class numbers of quadratic fields, Symp. Mat. 15 (1975), 403-411.Google Scholar
6. Hasse, Helmut, Ober die Klassenzahl des Körpers P(√-p) mit einer Primzahl p ≡ 1 (mod 23), Aequationes Math. 3 (1969), 165-169.Google Scholar
7. Hasse, Helmut, Über die Klassenzahl des Körpers P(√-2p) mit einer Primzahl p ≠ 2, J. Number Theory 1 (1969), 231-234.Google Scholar
8. Hasse, Helmut, Über die Teilbarkeit durch 23 der Klassenzahl imaginär-quadratischer Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilem, J. Reine Angew. Math. 241 (1970), 1-6.Google Scholar
9. Kaplan, Pierre, Divisibilité par 8 du nombre des classes des corps quadratiques dont le 2-groupe des classes est cyclique, et réciprocité biquadratique, J. Math. Soc. Japan 25 (1973), 596-608.Google Scholar
10. Kaplan, Pierre, Cycles d'ordre au moins 16 dans le 2-groupe des classes d'idéaux de certains corps quadratiques, Bull. Soc. Math. France Mém. No. 49-50 (1977), 113-124.Google Scholar
11. Kaplan, Pierre and Williams, Kenneth S., On the class numbers of Q(√?2p) modulo 16, for p ≡ 1 (mod 8) a prime, Acta Arith. (to appear).Google Scholar
12. Williams, Kenneth S., On the class number of Q(√-p) modulo 16, for p ≡ 1 (mod 8) a prime, Acta Arith. (to appear).Google Scholar