Geodesic
Arithmetic of certain~$\ell$-extensions ramified at three places.~III
Izvestiya. Mathematics , Tome 86 (2022) no. 6, pp. 1143-1161

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Let $\ell$ be a regular odd prime, $K$ the $\ell$ th cyclotomic field and $K=k(\sqrt[\ell]{a}\,)$, where $a$ is a positive integer. Under the assumption that there are exactly three places ramified in the extension $K_\infty/k_\infty$, we study the $\ell$-component of the class group of the field $K$. We prove that in the case $\ell>3$ there always is an unramified extension $\mathcal{N}/K$ such that $G(\mathcal{N}/K)\cong (\mathbb Z/\ell\mathbb Z)^2$ and all places over $\ell$ split completely in the extension $\mathcal{N}/K$. In the case $\ell=3$ we give a complete description of the situation. Some other results are obtained.
Keywords: Iwasawa theory, Tate module, extensions with restricted ramification. 
@article{IM2_2022_86_6_a5,
     author = {L. V. Kuz'min},
     title = {Arithmetic of certain~$\ell$-extensions ramified at three {places.~III}},
     journal = {Izvestiya. Mathematics },
     pages = {1143--1161},
     publisher = {mathdoc},
     volume = {86},
     number = {6},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_6_a5/}
}
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L. V. Kuz'min. Arithmetic of certain~$\ell$-extensions ramified at three places.~III. Izvestiya. Mathematics , Tome 86 (2022) no. 6, pp. 1143-1161. http://geodesic.mathdoc.fr/item/IM2_2022_86_6_a5/