Hopf-Galois module structure of tame biquadratic extensions

Paul J. Truman[1]

  • [1] School of Computing and Mathematics Keele University, ST5 5BG, UK

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 1, page 173-199
  • ISSN: 1246-7405

Abstract

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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 G ≅ C 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 = ℚ , and construct extensions exhibiting a variety of global behaviour, which implies that the direct analogue of the Hilbert-Speiser theorem does not hold.

How to cite

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Truman, Paul J.. "Hopf-Galois module structure of tame biquadratic extensions." Journal de Théorie des Nombres de Bordeaux 24.1 (2012): 173-199. <http://eudml.org/doc/251057>.

@article{Truman2012,
abstract = {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 $ G \cong C_\{2\} \times C_\{2\} $ and study in detail the local and global structure of the ring of integers $ \{\mathfrak\{O\}\}_\{L\}$ as a module over its associated order $ \{\mathfrak\{A\}\}_\{H\}$ in each of the Hopf algebras $ H $ giving a nonclassical Hopf-Galois structure on the extension. The results of [14] imply that $ \{\mathfrak\{O\}\}_\{L\}$ is locally free over each $ \{\mathfrak\{A\}\}_\{H\}$, and we derive necessary and sufficient conditions for $ \{\mathfrak\{O\}\}_\{L\}$ to be free over each $ \{\mathfrak\{A\}\}_\{H\}$. In particular, we consider the case $ K=\mathbb\{Q\}$, and construct extensions exhibiting a variety of global behaviour, which implies that the direct analogue of the Hilbert-Speiser theorem does not hold.},
affiliation = {School of Computing and Mathematics Keele University, ST5 5BG, UK},
author = {Truman, Paul J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {3},
number = {1},
pages = {173-199},
publisher = {Société Arithmétique de Bordeaux},
title = {Hopf-Galois module structure of tame biquadratic extensions},
url = {http://eudml.org/doc/251057},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Truman, Paul J.
TI - Hopf-Galois module structure of tame biquadratic extensions
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/3//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 1
SP - 173
EP - 199
AB - 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 $ G \cong C_{2} \times C_{2} $ and study in detail the local and global structure of the ring of integers $ {\mathfrak{O}}_{L}$ as a module over its associated order $ {\mathfrak{A}}_{H}$ in each of the Hopf algebras $ H $ giving a nonclassical Hopf-Galois structure on the extension. The results of [14] imply that $ {\mathfrak{O}}_{L}$ is locally free over each $ {\mathfrak{A}}_{H}$, and we derive necessary and sufficient conditions for $ {\mathfrak{O}}_{L}$ to be free over each $ {\mathfrak{A}}_{H}$. In particular, we consider the case $ K=\mathbb{Q}$, and construct extensions exhibiting a variety of global behaviour, which implies that the direct analogue of the Hilbert-Speiser theorem does not hold.
LA - eng
UR - http://eudml.org/doc/251057
ER -

References

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