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A congruence relating the class numbers of complex quadratic fields. (English) Zbl 0557.12003
Let \(n\) denote a positive integer and let \(p_ 1,\ldots,p_ s\) be \(s\) \((\geq 0)\) distinct primes \(\equiv 1\pmod 4\) and \(q_{s+1},\ldots,q_ n\) be \(n-s\) \((\geq 0)\) distinct primes \(\equiv 3\pmod 4\). Set \(d=(-1)^{n-s} p_ 1\cdots p_ s q_{s+1}\cdots q_ n\equiv 1\pmod 4\). The integer \(d\) is the discriminant of the quadratic field \(\mathbb Q(\sqrt{d})\), which is real or complex according as \(n-s\) is even or odd. The class number of the quadratic field of discriminant \(D\) is denoted by \(h(D)\).
In this paper a congruence of the type
\[ \begin{aligned} &\sum_{e| d,\;e>0,\;e\equiv 1\pmod 4}(c_ 1(d,e) h(-4e)+c_ 2(d,e) h(-8e))\\ &+\sum_{e| d,\;e<0,\;e\equiv 1\pmod 4}(c_ 3(d,e) h(e)+c_ 4(d,e) h(8e))\equiv c(d)\pmod{2^{n+2}}\end{aligned} \] is proved, where \(c_ i(d,e)\) \((i=1,\ldots,4)\) and \(c(d)\) are certain integers. When \(n=1\) this congruence reduces to well-known congruences modulo 8 due to Dirichlet and when \(n=2\) it gives a unified congruence for the 18 congruences modulo 16 proved by A. Pizer [J. Number Theory 8, 184–192 (1976; Zbl 0329.12003)].

11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
Zbl 0329.12003
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