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

Authors : Fouad Elmouhib, Mohamed Talbi, Abdelmalek Azizi

Pages : 20-31

Doi:10.24330/ieja.1388822

View : 73 | Download : 181

Publication Date : 2024-01-09

Article Type : Research Paper

Abstract :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.
Keywords : Pure metacyclic field, 5 class group, Hilbert 5 class field, capitulation

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