The maximal unramified extensions of certain complex Abelian number fields
Publications Mathématiques de Besançon (2015), pp. 93-104.

Nous combinons les minorations des discriminants avec des considérations portant sur la ramification pour montrer, inconditionnellement, que le corps Q(-7,61) n’a pas d’extension non-ramifiée non-triviale (ce résultat a été montré par Yamamura avec l’aide de GRH). Cela rend inconditionnelle la détermination des extensions non-ramifiées maximales des coprs quadratiques complexes de nombre de classes 2. Sous GRH, nous montrons un résultat analogue pour le sous-corps de degré 14 de Q(ζ 49 ) (corps non étudié même sous GRH).

We combine root discriminant bounds with a ramification argument to show unconditionally that Q(-7,61) has no nontrivial unramified extension, a result first proved by Yamamura under the generalized Riemann hypothesis (GRH). This renders unconditional his determination of the maximal unramified extensions of the complex quadratic fields with class number 2. Assuming the GRH, we prove an analogous result for the degree 14 subfield of the cyclotomic field Q(ζ 49 ), a case previously not handled by conditional root discriminant bounds alone.

Reçu le : 2014-11-05
Publié le : 2016-01-24
DOI : https://doi.org/10.5802/pmb.14
Classification : 11R20,  11R21,  11R29,  20E22
Mots clés: Abelian fields, group extensions, root discriminants, unramified extensions
@article{PMB_2015____93_0,
     author = {Siman Wong},
     title = {The maximal unramified extensions of certain complex Abelian number fields},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2015},
     pages = {93-104},
     doi = {10.5802/pmb.14},
     zbl = {1414.11137},
     language = {en},
     url = {pmb.centre-mersenne.org/item/PMB_2015____93_0/}
}
Siman Wong. The maximal unramified extensions of certain complex Abelian number fields. Publications Mathématiques de Besançon (2015), pp. 93-104. doi : 10.5802/pmb.14. https://pmb.centre-mersenne.org/item/PMB_2015____93_0/

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