Steinitz classes of some abelian and nonabelian extensions of even degree

Alessandro Cobbe

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 607-628
  • ISSN: 1246-7405

Abstract

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The Steinitz class of a number field extension K / k is an ideal class in the ring of integers 𝒪 k of k , which, together with the degree [ K : k ] of the extension determines the 𝒪 k -module structure of 𝒪 K . We denote by R t ( k , G ) the set of classes which are Steinitz classes of a tamely ramified G -extension of k . We will say that those classes are realizable for the group G ; it is conjectured that the set of realizable classes is always a group.In this paper we will develop some of the ideas contained in [7] to obtain some results in the case of groups of even order. In particular we show that to study the realizable Steinitz classes for abelian groups, it is enough to consider the case of cyclic groups of 2 -power degree.

How to cite

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Cobbe, Alessandro. "Steinitz classes of some abelian and nonabelian extensions of even degree." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 607-628. <http://eudml.org/doc/116423>.

@article{Cobbe2010,
abstract = {The Steinitz class of a number field extension $K/k$ is an ideal class in the ring of integers $\mathcal\{O\}_k$ of $k$, which, together with the degree $[K:k]$ of the extension determines the $\mathcal\{O\}_k$-module structure of $\mathcal\{O\}_K$. We denote by $\mathrm\{R\}_t(k,G)$ the set of classes which are Steinitz classes of a tamely ramified $G$-extension of $k$. We will say that those classes are realizable for the group $G$; it is conjectured that the set of realizable classes is always a group.In this paper we will develop some of the ideas contained in [7] to obtain some results in the case of groups of even order. In particular we show that to study the realizable Steinitz classes for abelian groups, it is enough to consider the case of cyclic groups of $2$-power degree.},
author = {Cobbe, Alessandro},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Steinitz classes; realizable classes; tame extensions of number fields; class field theory},
language = {eng},
number = {3},
pages = {607-628},
publisher = {Université Bordeaux 1},
title = {Steinitz classes of some abelian and nonabelian extensions of even degree},
url = {http://eudml.org/doc/116423},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Cobbe, Alessandro
TI - Steinitz classes of some abelian and nonabelian extensions of even degree
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 607
EP - 628
AB - The Steinitz class of a number field extension $K/k$ is an ideal class in the ring of integers $\mathcal{O}_k$ of $k$, which, together with the degree $[K:k]$ of the extension determines the $\mathcal{O}_k$-module structure of $\mathcal{O}_K$. We denote by $\mathrm{R}_t(k,G)$ the set of classes which are Steinitz classes of a tamely ramified $G$-extension of $k$. We will say that those classes are realizable for the group $G$; it is conjectured that the set of realizable classes is always a group.In this paper we will develop some of the ideas contained in [7] to obtain some results in the case of groups of even order. In particular we show that to study the realizable Steinitz classes for abelian groups, it is enough to consider the case of cyclic groups of $2$-power degree.
LA - eng
KW - Steinitz classes; realizable classes; tame extensions of number fields; class field theory
UR - http://eudml.org/doc/116423
ER -

References

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