Abstract
Let K = \( k(\sqrt \theta ) \) be a real cyclic quartic field, k be its quadratic subfield and \( \tilde K = k(\sqrt { - \theta } ) \) be the corresponding imaginary quartic field. Denote the class numbers of K, k and \( \tilde K \) by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h − = h K /h K and \( \tilde h^ - = h_{\tilde K} /h_k \) are obtained via studying the p-adic L-functions of the fields.
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References
Ankeny N, Artin E, Chowla S. The class number of quadratic numberfields. Ann of Math, 52(2): 479–493 (1952)
Lu H. Congruences for the class number of quadratic fields. Abh Math Sen Univ Hamberg, 47: 254–258 (1983)
Feng K. Ankeny-Artin-Chowla formula on cubic cyclic number fields. J China Univ Sci Tech, 12: 20–27 (1982)
Zhang X. Congruences of class numbers of general cubic cyclic number fields. J China Univ Sci Tech, 17(2): 141–145 (1987)
Zhang X. Ten formulae of type Ankeny-Artin-Chowla for class numbers of general cyclic quartic fields. Sci China Ser A-Math, 32(4): 417–428 (1989)
Liu T. Congruence for the class numbers of real cyclic sextic number fields. Sci China Ser A-Math, 42(10): 1009–1018 (1999)
Zhang X. Congruence modulo 8 for the class numbers of general quadratic fields ℚ(√m) and ℚ(√−m). J Number Theory, 32(3): 332–338 (1989)
Hasse H. Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlk"orpern. Abh Deutsch Akad Wiss Berlin Math, 2: 1–95 (1948)
Lazarus A J. On the class number and unit index of simplest quartic fields. Nagoya Math J, 121: 1–13 (1991)
Feng K. Explicit description of quartic cyclic numberfields. Acta Math Sinica, 27(3): 410–424 (1984)
Zhang X. Introduction to Algebraic Number Theory. Changsha: Hunan Edu Publ, 1999
Washington L C. Introduction to Cyclotomic Fields. New York: Spring-Verlag, 1982
Kudo A. On a class number relation of imaginary abelian fields. J Math Soc Japan, 27(1): 150–159 (1975)
Shiratani K. Kummer’s congruence for generalized Bernoulli numbers and its application. Mem Fac Sci Kyushu Univ, 26: 119–138 (1972)
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This work was supported by National Natural Science Foundation of China (Grant No. 10771111)
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Ma, L., Li, W. & Zhang, X. Congruence formulae modulo powers of 2 for class numbers of cyclic quartic fields. Sci. China Ser. A-Math. 52, 417–426 (2009). https://doi.org/10.1007/s11425-009-0021-y
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DOI: https://doi.org/10.1007/s11425-009-0021-y