Geodesic
Arithmetic of Certain $\ell $-Extensions Ramified at Three Places
Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 78-99.

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Let $\ell $ be a regular odd prime number, $k$ the $\ell $th cyclotomic field, $k_\infty $ the cyclotomic $\mathbb Z_\ell $-extension of $k$, $K$ a cyclic extension of $k$ of degree $\ell $, and $K_\infty =K\cdot k_\infty $. Under the assumption that there are exactly three places not over $\ell $ that ramify in the extension $K_\infty /k_\infty $ and $K$ satisfies some additional conditions, we study the structure of the Iwasawa module $T_\ell (K_\infty )$ of $K_\infty $ as a Galois module. In particular, we prove that $T_\ell (K_\infty )$ is a cyclic $G(K_\infty /k_\infty )$-module and the Galois group $\Gamma =G(K_\infty /K)$ acts on $T_\ell (K_\infty )$ as $\sqrt {\varkappa }$, where $\varkappa \colon \Gamma \to \mathbb Z_\ell ^\times $ is the cyclotomic character. 
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     title = {Arithmetic of {Certain} $\ell ${-Extensions} {Ramified} at {Three} {Places}},
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L. V. Kuz'min. Arithmetic of Certain $\ell $-Extensions Ramified at Three Places. Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 78-99. http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a3/

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