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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Real quadratic fields with class numbers divisible by five


Author: Charles J. Parry
Journal: Math. Comp. 31 (1977), 1019-1029
MSC: Primary 12A25; Secondary 12A50
MathSciNet review: 0498483
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Abstract: Conditions are given for a real quadratic field to have class number divisible by five. If 5 does not divide m, then a necessary condition for 5 to divide the class number of the real quadratic field with conductor m or 5m is that 5 divide the class number of a certain cyclic biquadratic field with conductor 5m. Conversely, if 5 divides the class number of the cyclic field, then either one of the quadratic fields has class number divisible by 5 or one of their fundamental units satisfies a certain congruence condition modulo 25.


References [Enhancements On Off] (What's this?)

  • [1] Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239 (14,141a)
  • [2] E. HECKE, Vorlesungen über die Theorie der algebraischen Zahlen, Leipzig, 1923.
  • [3] C.S. HERZ, Construction of Class Fields, Lecture Notes in Math., vol. 21, Springer-Verlag, Berlin and New York, 1966. MR 34 #1278.
  • [4] E. L. INCE, Cycles of Reduced Ideals in Quadratic Fields, British Math. Assn. Tables, Vol. 4, London, 1934.
  • [5] Tomio Kubota, Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen biquadratischen Zahlkörpers, Nagoya Math. J. 6 (1953), 119–127 (German). MR 0059960 (15,605e)
  • [6] Sigekatu Kuroda, Über den Dirichletschen Körper, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1943), 383–406 (German). MR 0021031 (9,12f)
  • [7] Charles J. Parry, Units of algebraic numberfields, J. Number Theory 7 (1975), no. 4, 385–388. MR 0384752 (52 #5625)
  • [8] C. WALTER, Class Number Relations in Algebraic Number Fields, Ph.D. Thesis, University of Cambridge, 1976.
  • [9] P. J. Weinberger, Real quadratic fields with class numbers divisible by 𝑛, J. Number Theory 5 (1973), 237–241. MR 0335471 (49 #252)
  • [10] Yoshihiko Yamamoto, On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 57–76. MR 0266898 (42 #1800)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1977-0498483-X
PII: S 0025-5718(1977)0498483-X
Article copyright: © Copyright 1977 American Mathematical Society