Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. Byott[1]; Bouchaïb Sodaïgui[2]

  • [1] College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, UK
  • [2] Département de Mathématiques, Université de Valenciennes, Le Mont Houy, 59313 Valenciennes Cedex 9, France

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 1, page 303-371
  • ISSN: 0373-0956

Abstract

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Given an algebraic number field k and a finite group Γ , we write ℛ ( O k [ Γ ] ) for the subset of the locally free classgroup Cl ( O k [ Γ ] ) consisting of the classes of rings of integers O N in tame Galois extensions N / k with Gal ( N / k ) ≅ Γ . We determine ℛ ( O k [ Γ ] ) , and show it is a subgroup of Cl ( O k [ Γ ] ) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ = V ⋊ C , where V is an elementary abelian group of order p r and C is a cyclic group of order m > 1 acting faithfully on V and making V into an irreducible 𝔽 p [ C ] -module. This extends and refines results of Byott, Greither and Sodaïgui for p = 2 in Crelle, respectively of Bruche and Sodaïgui for p > 2 in J. Number Theory, which cover only the case m = p r - 1 and determine only the image ℛ ( ℳ ) of ℛ ( O k [ Γ ] ) under extension of scalars from O k [ Γ ] to a maximal order ℳ ⊃ O k [ Γ ] in k [ Γ ] . The main result here thus generalizes the calculation of ℛ ( O k [ A 4 ] ) for the alternating group A 4 of degree 4 (the case p = r = 2 ) given by Byott and Sodaïgui in Compositio.

How to cite

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Byott, Nigel P., and Sodaïgui, Bouchaïb. "Realizable Galois module classes over the group ring for non abelian extensions." Annales de l’institut Fourier 63.1 (2013): 303-371. <http://eudml.org/doc/275530>.

@article{Byott2013,
abstract = {Given an algebraic number field $k$ and a finite group $\Gamma $, we write $\mathcal\{R\}(O_k[\Gamma ])$ for the subset of the locally free classgroup $\mathrm\{Cl\}(O_k[\Gamma ])$ consisting of the classes of rings of integers $O_N$ in tame Galois extensions $N/k$ with $\mathrm\{Gal\}(N/k) \cong \Gamma $. We determine $\mathcal\{R\}(O_k[\Gamma ])$, and show it is a subgroup of $\mathrm\{Cl\}(O_k[\Gamma ])$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $\Gamma =V \rtimes C$, where $V$ is an elementary abelian group of order $p^r$ and $C$ is a cyclic group of order $m&gt;1$ acting faithfully on $V$ and making $V$ into an irreducible $\mathbb\{F\}_p[C]$-module. This extends and refines results of Byott, Greither and Sodaïgui for $p=2$ in Crelle, respectively of Bruche and Sodaïgui for $p&gt;2$ in J. Number Theory, which cover only the case $m=p^r-1$ and determine only the image $\mathcal\{R\}(\mathcal\{M\})$ of $\mathcal\{R\}(O_k[\Gamma ])$ under extension of scalars from $O_k[\Gamma ]$ to a maximal order $\mathcal\{M\} \supset O_k[\Gamma ]$ in $k[\Gamma ]$. The main result here thus generalizes the calculation of $\mathcal\{R\}(O_k[A_4])$ for the alternating group $A_4$ of degree 4 (the case $p=r=2$) given by Byott and Sodaïgui in Compositio.},
affiliation = {College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, UK; Département de Mathématiques, Université de Valenciennes, Le Mont Houy, 59313 Valenciennes Cedex 9, France},
author = {Byott, Nigel P., Sodaïgui, Bouchaïb},
journal = {Annales de l’institut Fourier},
keywords = {Galois module structure; Rings of algebraic integers; Locally free classgroup; Fröhlich-Lagrange resolvent; Realizable classes; Embedding problem; Stickelberger ideal; Cyclic codes; rings of algebraic integers; locally free class group; realizable classes; embedding problem; cyclic codes},
language = {eng},
number = {1},
pages = {303-371},
publisher = {Association des Annales de l’institut Fourier},
title = {Realizable Galois module classes over the group ring for non abelian extensions},
url = {http://eudml.org/doc/275530},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Byott, Nigel P.
AU - Sodaïgui, Bouchaïb
TI - Realizable Galois module classes over the group ring for non abelian extensions
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 1
SP - 303
EP - 371
AB - Given an algebraic number field $k$ and a finite group $\Gamma $, we write $\mathcal{R}(O_k[\Gamma ])$ for the subset of the locally free classgroup $\mathrm{Cl}(O_k[\Gamma ])$ consisting of the classes of rings of integers $O_N$ in tame Galois extensions $N/k$ with $\mathrm{Gal}(N/k) \cong \Gamma $. We determine $\mathcal{R}(O_k[\Gamma ])$, and show it is a subgroup of $\mathrm{Cl}(O_k[\Gamma ])$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $\Gamma =V \rtimes C$, where $V$ is an elementary abelian group of order $p^r$ and $C$ is a cyclic group of order $m&gt;1$ acting faithfully on $V$ and making $V$ into an irreducible $\mathbb{F}_p[C]$-module. This extends and refines results of Byott, Greither and Sodaïgui for $p=2$ in Crelle, respectively of Bruche and Sodaïgui for $p&gt;2$ in J. Number Theory, which cover only the case $m=p^r-1$ and determine only the image $\mathcal{R}(\mathcal{M})$ of $\mathcal{R}(O_k[\Gamma ])$ under extension of scalars from $O_k[\Gamma ]$ to a maximal order $\mathcal{M} \supset O_k[\Gamma ]$ in $k[\Gamma ]$. The main result here thus generalizes the calculation of $\mathcal{R}(O_k[A_4])$ for the alternating group $A_4$ of degree 4 (the case $p=r=2$) given by Byott and Sodaïgui in Compositio.
LA - eng
KW - Galois module structure; Rings of algebraic integers; Locally free classgroup; Fröhlich-Lagrange resolvent; Realizable classes; Embedding problem; Stickelberger ideal; Cyclic codes; rings of algebraic integers; locally free class group; realizable classes; embedding problem; cyclic codes
UR - http://eudml.org/doc/275530
ER -

References

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