Real quadratic fields with class numbers divisible by five

Parry, Charles J.

doi:10.1090/S0025-5718-1977-0498483-X

Real quadratic fields with class numbers divisible by five
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by Charles J. Parry PDF

Math. Comp. 31 (1977), 1019-1029 Request permission

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

Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239

Vorlesungen über die Theorie der algebraischen Zahlen

Construction of Class Fields

34

Cycles of Reduced Ideals in Quadratic Fields

Tomio Kubota, Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen biquadratischen Zahlkörpers, Nagoya Math. J. 6 (1953), 119–127 (German). MR 59960
Sigekatu Kuroda, Über den Dirichletschen Körper, J. Fac. Sci. Imp. Univ. Tokyo Sect. I. 4 (1943), 383–406 (German). MR 0021031
Charles J. Parry, Units of algebraic numberfields, J. Number Theory 7 (1975), no. 4, 385–388. MR 384752, DOI 10.1016/0022-314X(75)90042-6

Class Number Relations in Algebraic Number Fields

P. J. Weinberger, Real quadratic fields with class numbers divisible by $n$, J. Number Theory 5 (1973), 237–241. MR 335471, DOI 10.1016/0022-314X(73)90049-8
Yoshihiko Yamamoto, On unramified Galois extensions of quadratic number fields, Osaka Math. J. 7 (1970), 57–76. MR 266898