Normal integral basis of an unramified quadratic extension over a cyclotomic $\mathbb{Z}_2$-extension

Humio Ichimura; Hiroki Sumida-Takahashi

doi:10.5802/jtnb.942

Normal integral basis of an unramified quadratic extension over a cyclotomic

ℤ_{2}

-extension

Humio Ichimura ¹; Hiroki Sumida-Takahashi ²

¹ Faculty of Science Ibaraki University Bunkyo 2-1-1, Mito, 310-8512, Japan
² Faculty of Engineering Tokushima University 2-1 Minami-josanjima-cho, Tokushima, 770-8506, Japan

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 325-345.

Abstract
Résumé

Let $ℓ$ be an odd prime number. Let $K / ℚ$ be a real cyclic extension of degree $ℓ$ , $A_{K}$ the $2$ -part of the ideal class group of $K$ , and $H / K$ the class field corresponding to $A_{K} / A_{K}^{2}$ . Let $K_{n}$ be the $n$ th layer of the cyclotomic $ℤ_{2}$ -extension over $K$ . We consider the questions (Q1) “does $H / K$ has a normal integral basis?”, and (Q2) “if not, does the pushed-up extension $H K_{n} / K_{n}$ has a normal integral basis for some $n \geq 1$ ?” Under some assumptions on $ℓ$ and $K$ , we answer these questions in terms of the $2$ -adic $L$ -function associated to the base field $K$ . We also give some numerical examples.

Soit $ℓ$ un nombre premier impair. Soient $K / ℚ$ une extension cyclique réelle de degré $ℓ$ , $A_{K}$ la $2$ -partie du groupe des classes d’idéaux de $K$ , et $H / K$ le corps des classes correspondant à $A_{K} / A_{K}^{2}$ . Soit $K_{n}$ la $n$ -ème couche de la $ℤ_{2}$ -extension cyclotomique sur $K$ . Nous considérons les questions (Q1) “existe-il une base intégrale normale pour $H / K$ ?” et (Q2) “sinon, l’extension induite $H K_{n} / K_{n}$ a-t-elle une base intégrale normale pour un certain $n \geq 1$ ?” Sous quelques hypothèses sur $ℓ$ et $K$ , nous répondrons à ces questions en termes de la fonction $L$ $2$ -adique associée au corps $K$ de base. De plus, nous donnons quelques exemples numériques.

Received: 2014-02-27
Revised: 2014-12-19
Accepted: 2015-02-16
Published online: 2017-01-02

Zbl

DOI: 10.5802/jtnb.942

Classification: 11R33, 11R23
Keywords: Normal integral basis, unramified quadratic extension, cyclotomic $\mathbb{Z}_2$-extension.

Author's affiliations:

Humio Ichimura ¹; Hiroki Sumida-Takahashi ²

¹ Faculty of Science Ibaraki University Bunkyo 2-1-1, Mito, 310-8512, Japan
² Faculty of Engineering Tokushima University 2-1 Minami-josanjima-cho, Tokushima, 770-8506, Japan

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     author = {Humio Ichimura and Hiroki Sumida-Takahashi},
     title = {Normal integral basis of an unramified quadratic extension over a cyclotomic $\mathbb{Z}_2$-extension},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {325--345},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {28},
     number = {2},
     year = {2016},
     doi = {10.5802/jtnb.942},
     zbl = {1358.11120},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.942/}
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Humio Ichimura; Hiroki Sumida-Takahashi. Normal integral basis of an unramified quadratic extension over a cyclotomic $\mathbb{Z}_2$-extension. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 325-345. doi : 10.5802/jtnb.942. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.942/

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