Turkish Journal of Mathematics
DOI
10.3906/mat-2103-58
Abstract
Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a primitive cube root of unity. Then $k=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is a pure metacyclic field with group $\mathrm{Gal}(k/\mathbb{Q})\simeq S_3$. In the case that $k$ possesses a $3$-class group $C_{k,3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $k$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $k_3^{(\infty)}$, that is the $3$-class field tower of $k$, are drawn.
Keywords
Maximal unramified pro-$3$-extension, capitulation, Galois action, pure metacyclic $S_3$-fields, pure cubic fields, finite $3$-groups, descendant trees, presentations, relation rank, $p$-group generation algorithm
First Page
2225
Last Page
2247
Recommended Citation
AOUISSI, SIHAM; TALBI, MOHAMED; MAYER, DANIEL C.; and ISMAILI, MOULAY CHRIF
(2021)
"3-Principalization over S_3-fields,"
Turkish Journal of Mathematics: Vol. 45:
No.
5, Article 24.
https://doi.org/10.3906/mat-2103-58
Available at:
https://journals.tubitak.gov.tr/math/vol45/iss5/24
