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Algèbre et Théorie des Nombres
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Siman Wong
The maximal unramified extensions of certain complex Abelian number fields
Publications mathématiques de Besançon (2015), p. 93-104, doi: 10.5802/pmb.14
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Class. Math.: 11R20, 11R21, 11R29, 20E22
Mots clés: Abelian fields, group extensions, root discriminants, unramified extensions

Résumé

Nous combinons les minorations des discriminants avec des considérations portant sur la ramification pour montrer, inconditionnellement, que le corps $ {\mathbf{Q}}(\sqrt{-7},\sqrt{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 $ {\mathbf{Q}}(\zeta _{49}) $ (corps non étudié même sous GRH).

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