Geodesic
On a family of algebraic number fields with finite 3-class field tower
Sbornik. Mathematics, Tome 215 (2024) no. 7, pp. 911-919 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\ell=3$, $k=\mathbb Q(\sqrt{-3})$ and $K=k(\sqrt[3]{a})$, where $a$ is a natural number such that $a^2\equiv 1\pmod 9$. Under the assumption that there are exactly three places not over $\ell$ that ramify in the extension $K_\infty/k_\infty$, where $k_\infty$ and $K_\infty$ are cyclotomic $\mathbb Z_3$-extensions of the fields $k$ and $K$, respectively, we study 3-class field towers for intermediate fields $K_n$ of the extension $K_\infty/K$. It is shown that for each $K_n$ the 3-class field tower of the field $K_n$ terminates already at the first step, which means that the Galois group of the extension $\mathbf H_\ell(K_n)/K_n$, where $\mathbf H_\ell(K_n)$ is the maximal unramified $\ell$-extension of the field $K_n$, is Abelian. Bibliography: 7 titles.
Keywords: Iwasawa theory, Tate module, extensions with bounded ramification, Riemann–Hurwitz formula, class field tower. 
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     title = {On a~family of algebraic number fields with finite 3-class field tower},
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L. V. Kuz'min. On a family of algebraic number fields with finite 3-class field tower. Sbornik. Mathematics, Tome 215 (2024) no. 7, pp. 911-919. http://geodesic.mathdoc.fr/item/SM_2024_215_7_a1/

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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 | MR | Zbl

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

[4] L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places. IV”, Izv. Math., 88:2 (2024), 270–283 | DOI | MR | Zbl

[5] L. V. Kuz'min, “An analog of the Riemann–Hurwitz formula for one type of $l$-extension of algebraic number fields”, Math. USSR-Izv., 36:2 (1991), 325–347 | DOI | MR | Zbl

[6] L. V. Kuz'min, “New explicit formulas for the norm residue symbol, and their applications”, Math. USSR-Izv., 37:3 (1991), 555–586 | DOI | MR | Zbl

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