Geodesic
Arithmetic of certain $\ell$-extensions ramified at three places.~IV
Izvestiya. Mathematics , Tome 88 (2024) no. 2, pp. 270-283.

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Let $\ell$ be an odd regular prime, $k$ be the $\ell$th cyclotomic field, and $K=k(\sqrt[\ell]{a})$, where $a$ is a natural number that has exactly three distinct prime divisors. Assuming that there are exactly three places ramified in $K_\infty/k_\infty$, we study the $\ell$-component of the class group of the field $K$. For $\ell>3$, we prove that there always exists an unramified extension $\mathcal{N}/K$ such that $G(\mathcal{N}/K)\cong (\mathbb Z/\ell\mathbb Z)^3$, and all places over $\ell$ split completely in $\mathcal{N}/K$. If $\ell=3$ and $a$ is of the form $a=p^rq^s$, we give a complete description of the possible structure of the $\ell$-component of the class group of $K$. Some other results are also obtained.
Keywords: Iwasawa theory, Tate module, extension with given ramification, the Riemann–Hurwitz formula. 
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     title = {Arithmetic of certain $\ell$-extensions ramified at three {places.~IV}},
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L. V. Kuz'min. Arithmetic of certain $\ell$-extensions ramified at three places.~IV. Izvestiya. Mathematics , Tome 88 (2024) no. 2, pp. 270-283. http://geodesic.mathdoc.fr/item/IM2_2024_88_2_a4/

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[2] L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places. II”, Izv. Math., 85:5 (2021), 953–971 | DOI

[3] L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places. III”, Izv. Math., 86:6 (2022), 1143–1161 | DOI

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[6] L. V. Kuz'min, “The Tate module for algebraic number fields”, Math. USSR-Izv., 6:2 (1972), 263–321 | DOI