Galois module structure of rings of integers

Martin J. Taylor

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 3, page 11-48
  • ISSN: 0373-0956

Abstract

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Let E / F be a Galois extension of number fields with Γ = Gal ( E / F ) and with property that the divisors of ( E : F ) are non-ramified in E / Q . We denote the ring of integers of E by 𝒪 E and we study 𝒪 E as a Z Γ -module. In particular we show that the fourth power of the (locally free) class of 𝒪 E is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of ℰ E , together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.

How to cite

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Taylor, Martin J.. "Galois module structure of rings of integers." Annales de l'institut Fourier 30.3 (1980): 11-48. <http://eudml.org/doc/74455>.

@article{Taylor1980,
abstract = {Let $E/F$ be a Galois extension of number fields with $\Gamma =$ Gal$(E/F)$ and with property that the divisors of $(E:F)$ are non-ramified in $E/\{\bf Q\}$. We denote the ring of integers of $E$ by $\{\cal O\}_E$ and we study $\{\cal O\}_E$ as a $\{\bf Z\}\Gamma $-module. In particular we show that the fourth power of the (locally free) class of $\{\cal O\}_E$ is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of $\{\cal E\}_E$, together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.},
author = {Taylor, Martin J.},
journal = {Annales de l'institut Fourier},
keywords = {Gauß sums; Galois extension; ring of integers; class groups of modules},
language = {eng},
number = {3},
pages = {11-48},
publisher = {Association des Annales de l'Institut Fourier},
title = {Galois module structure of rings of integers},
url = {http://eudml.org/doc/74455},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Taylor, Martin J.
TI - Galois module structure of rings of integers
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 3
SP - 11
EP - 48
AB - Let $E/F$ be a Galois extension of number fields with $\Gamma =$ Gal$(E/F)$ and with property that the divisors of $(E:F)$ are non-ramified in $E/{\bf Q}$. We denote the ring of integers of $E$ by ${\cal O}_E$ and we study ${\cal O}_E$ as a ${\bf Z}\Gamma $-module. In particular we show that the fourth power of the (locally free) class of ${\cal O}_E$ is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of ${\cal E}_E$, together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.
LA - eng
KW - Gauß sums; Galois extension; ring of integers; class groups of modules
UR - http://eudml.org/doc/74455
ER -

References

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  1. [1] Ph. CASSOU-NOGUES, Quelques théorèmes de base normale d'entiers, Ann. Inst. Fourier, 28, 3 (1978), 1-33. Zbl0368.12004
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  14. [14] J. TATE, Local constants, Algebraic Number Fields, (Proc. Durham Symposium), Academic Press, London, 1977. 
  15. [15] M.J. TAYLOR, Galois module structure of relative abelian extensions, J. reine angew. Math., 303/4 (1978), 97-101. Zbl0384.12007MR80e:12006
  16. [16] M.J. TAYLOR, On the self-duality of a ring of integers as a Galois module, Inventiones Mathematicae, 46 (1978), 173-177. Zbl0381.12007MR57 #12460
  17. [17] M.J. TAYLOR, A logarithmic description of locally free classgroups of finite groups, to appear in the Journal of Algebra. 

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