A determinant concerning the relative class number of the cyclotomic field \(\mathbb{Q} (\zeta_{p^ n})\) (Q2785642)
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scientific article; zbMATH DE number 981786
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A determinant concerning the relative class number of the cyclotomic field \(\mathbb{Q} (\zeta_{p^ n})\) |
scientific article; zbMATH DE number 981786 |
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10 August 1997
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relative class number
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cyclotomic number field
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determinant
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minus class number
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A determinant concerning the relative class number of the cyclotomic field \(\mathbb{Q} (\zeta_{p^ n})\) (English)
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It is well-known that the minus class number of the cyclotomic number field \(\mathbb{Q}(\zeta_p)\) can be expressed as a product of two factors, one of which is easy to compute, and the other being a determinant of a matrix with a simple structure; the best known example is Maillet's matrix.NEWLINENEWLINENEWLINEIn this paper, the author derives a similar formula for the minus class number of the \(p^n\)th roots of unity. By applying Hadamard's theorem to his result, he gets an upper bound for the minus class number, which generalizes an estimate of \textit{K. Feng} [Proc. Am. Math. Soc. 84, 479-482 (1982; Zbl 0493.12009)] in the case \(n=1\).
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