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A congruence for the second factor of the class number of a cyclotomic field. (English) Zbl 0177.07501

Let \(\zeta =e^{2\pi i/p}\), where \(p = 2m+1\) is a prime \(>3\). Put \(K=O(\zeta)\), the cyclotomic field generated by \(\zeta\). If \(h\) denotes the class number of \(K\), then it is familiar that \(h=h_1h_2\), where \(h_1\) is the first factor and \(h_2\) is the second factor of the class number. It is well known that \(h\) is divisible by \(p\) if and only if \(h_1\) is divisible by \(p\); this is equivalent to the statement \(p\mid h_2\to p\mid h_1\). In the present paper the congruence \(h_2G\equiv \pm h_1\pmod p\) is proved; \(G\) is a rational integer depending only on \(p\). Indeed, \[ G \equiv (-1)^{n+1} 2^{n+3} G_0^{-1}C \pmod p,\]
where \(G_0\) is the difference product of the quadratic residues \(\ne 1\) of \(p\). To define \(C\), let \(g\) denote a primitive root (mod \(p)\) and \(\varepsilon_1(\zeta),\ldots,\varepsilon_{n-1}(\zeta)\) a fundamental system of units of \(K\). Put
\[ \zeta \frac{\varepsilon '_k(\zeta)}{\varepsilon_k(\zeta)} = \sum_{s=0}^{p-2}c_{k,s}\zeta^{g^s}\qquad (k=1,\ldots,m-1), \] \[ C_{k,n} = \sum_{s=0}^{p-2}c_{k,s} g^{(2n-1)s}\qquad (k,n=1,\ldots,m-1). \]
Then \(C = \det(C_{kn})\) \((k,n =1, \ldots,m-1)\).
(It should be noted that \(G_0\) is incorrectly defined in the paper.)

MSC:

11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
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