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

Year 2024, Volume: 35 Issue: 35, 20 - 31, 09.01.2024

Fouad Elmouhib Mohamed Talbi Abdelmalek Azizi

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

Cited By: 1

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

References

• 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.
• E. Hecke, Lectures on the Theory of Algebraic Numbers, Graduate Texts in Mathematics, 77, Springer-Verlag, New York-Berlin, 1981.\label{Hec}
• M. Kulkarni, D. Majumdar and B. Sury, $l$-class groups of cyclic extension of prime degree $l$, J. Ramanujan Math. Soc., 30(4) (2015), 413-454.\label{Mani}
• L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New-York, 1982.\label{washint}
• The PARI Group, PARI/GP, Version 2.4.9, Bordeaux, 2017, http://pari.math.u-bordeaux.fr\label{PARI}