Abstract
Let \(k \,=\, \mathbb {Q}(\root 5 \of {n},\zeta _5)\), where n is a positive integer 5-th power-free, whose 5-class group denoted by \(C_{k,5}\) is isomorphic to \(\mathbb {Z}/5\mathbb {Z}\times \mathbb {Z}/5\mathbb {Z}\). Let \(k_0\,=\,\mathbb {Q}(\zeta _5)\) be the cyclotomic field containing a primitive 5-th root of unity \(\zeta _5\). Let \(C_{k,5}^{(\sigma )}\) be the group of ambiguous classes under the action of \(\mathrm{Gal}(k/k_0)\) = \(\langle \sigma \rangle \). The aim of this paper is to determine all naturals n such that the group of ambiguous classes \(C_{k,5}^{(\sigma )}\) has rank 1 or 2.
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Communicated by B Sury.
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Azizi, A., Elmouhib, F. & Talbi, M. 5-Rank of ambiguous class groups of quintic Kummer extensions. Proc Math Sci 132, 12 (2022). https://doi.org/10.1007/s12044-022-00660-z
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DOI: https://doi.org/10.1007/s12044-022-00660-z
Keywords
- Pure quintic fields
- quintic Kummer extensions
- cyclotomic fields
- 5-group of ambiguous ideal classes
- 5-rank