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Real Quadratic Fields With Class Numbers Divisible by Five

Charles J. Parry
Mathematics of Computation
Vol. 31, No. 140 (Oct., 1977), pp. 1019-1029
DOI: 10.2307/2006134
Stable URL: http://www.jstor.org/stable/2006134
Page Count: 11
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Real Quadratic Fields With Class Numbers Divisible by Five
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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.

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