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The relative class numbers of imaginary cyclic fields of degrees 4, 6, 8, and 10. (English) Zbl 0787.11046
For an imaginary abelian field \(K\), let \(n\) be the conductor and \(h^ -\) the relative class number of \(K\). Let \(K^ +\) denote the maximal real subfield of \(K\). The author derives a formula for \(h^ -\) in terms of a generalized Maillet determinant, provided no prime factor of \(n\) splits in \(K/K^ +\). He then specializes to \(n = p\), an odd prime, and writes down the determinant involved given that \(K\) is of degree 4, 6, 8, or 10. In these cases he applies the formula to compute \(h^ -\) for \(p < 500,000\). A supplement displays the results for the degrees 6, 8, and 10 up to \(p < 10,000\).

MSC:
11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
11-04 Software, source code, etc. for problems pertaining to number theory
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