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. 
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     title = {Arithmetic of certain~$\ell$-extensions ramified at three {places.~III}},
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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/

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[2] L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places. II”, Izv. Ross. Akad. Nauk Ser. Mat., 85:5 (2021), 132–151 ; English transl. Izv. Math., 85:5 (2021), 953–971 | DOI | MR | Zbl | DOI

[3] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ, 1956 ; Russian transl. Inostr. Lit., Moscow, 1960 | MR | Zbl | MR