Hopf-Galois module structure of tame biquadratic extensions
Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 173-199.

In [14] we studied the nonclassical Hopf-Galois module structure of rings of algebraic integers in some tamely ramified extensions of local and global fields, and proved a partial generalisation of Noether’s theorem to this setting. In this paper we consider tame Galois extensions of number fields L/K with group GC 2 ×C 2 and study in detail the local and global structure of the ring of integers 𝔒 L as a module over its associated order 𝔄 H in each of the Hopf algebras H giving a nonclassical Hopf-Galois structure on the extension. The results of [14] imply that 𝔒 L is locally free over each 𝔄 H , and we derive necessary and sufficient conditions for 𝔒 L to be free over each 𝔄 H . In particular, we consider the case K=, and construct extensions exhibiting a variety of global behaviour, which implies that the direct analogue of the Hilbert-Speiser theorem does not hold.

Dans [14], nous avons étudié la structure de Hopf-Galois module non classique des anneaux d’entiers dans des extensions modérément ramifiées de corps locaux et globaux, et avons prouvé une généralisation partielle du théorème de Noether dans ce contexte. Dans le présent article, nous considérons des extensions galoisiennes modérées de corps de nombres L/K de groupe GC 2 ×C 2 et étudions en détail la structure locale et globale de l’anneau des entiers 𝔒 L comme module sur son ordre associé 𝔄 H dans chacune des algèbres de Hopf H donnant une structure de Hopf-Galois non classique sur l’extension. Les résultats de [14] impliquent que 𝔒 L est localement libre sur chaque 𝔄 H , et nous en tirons des conditions nécessaires et suffisantes pour que 𝔒 L soit libre sur chaque 𝔄 H . En particulier, nous considérons le cas K=, et construisons des extensions possédant une grande diversité de comportement global, ce qui implique que l’analogue direct du théorème d’Hilbert-Speiser n’est pas vrai.

Received:
Published online:
DOI: 10.5802/jtnb.792
Paul J. Truman 1

1 School of Computing and Mathematics Keele University, ST5 5BG, UK
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Paul J. Truman. Hopf-Galois module structure of tame biquadratic extensions. Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 173-199. doi : 10.5802/jtnb.792. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.792/

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