On the capitulation problem of some pure metacyclic fields of degree 20 II

Yıl 2024, , 20 - 31, 09.01.2024

Fouad Elmouhib Mohamed Talbi Abdelmalek Azizi

https://doi.org/10.24330/ieja.1388822

Cited By: 1

Öz

Let $n$ be a $5^{th}$ power-free natural number and $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $\zeta_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ is a pure metacyclic field of absolute degree $20$. In the case that $k$ possesses a $5$-class group $C_{k,5}$ of type $(5,5)$ and all the classes are ambiguous under the action of $Gal(k/k_0)$, the capitulation of $5$-ideal classes of $k$ in its unramified cyclic quintic extensions is determined.

Anahtar Kelimeler

Pure metacyclic field, 5-class group, Hilbert 5-class field, capitulation

Kaynakça

• A. Azizi, F. Elmouhib and M. Talbi, 5-rank of ambiguous class groups of quintic Kummer extensions, Proc. Indian Acad. Sci. Math. Sci., 132(12) (2022), 14 pp.
• F. Elmouhib, M. Talbi and A. Azizi, On the capitulation problem of some pure metacyclic fields of degree 20., Palest. J. Math., 11(1) (2022), 260-267.
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