Note on the Galois module structure of quadratic extensions

Günter Lettl

Colloquium Mathematicae (1994)

  • Volume: 67, Issue: 1, page 15-19
  • ISSN: 0010-1354

Abstract

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In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, ℚ ( p ) . For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of ℚ ( n ) / ℚ ( n ) + .

How to cite

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Lettl, Günter. "Note on the Galois module structure of quadratic extensions." Colloquium Mathematicae 67.1 (1994): 15-19. <http://eudml.org/doc/210258>.

@article{Lettl1994,
abstract = {In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, $ℚ^\{(p)\}$. For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of $ℚ^\{(n)\}/ℚ^\{(n)^+\}$.},
author = {Lettl, Günter},
journal = {Colloquium Mathematicae},
keywords = {associated order; relative extensions; quadratic extensions; ring of integers; Galois module structure of cyclotomic fields},
language = {eng},
number = {1},
pages = {15-19},
title = {Note on the Galois module structure of quadratic extensions},
url = {http://eudml.org/doc/210258},
volume = {67},
year = {1994},
}

TY - JOUR
AU - Lettl, Günter
TI - Note on the Galois module structure of quadratic extensions
JO - Colloquium Mathematicae
PY - 1994
VL - 67
IS - 1
SP - 15
EP - 19
AB - In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, $ℚ^{(p)}$. For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of $ℚ^{(n)}/ℚ^{(n)^+}$.
LA - eng
KW - associated order; relative extensions; quadratic extensions; ring of integers; Galois module structure of cyclotomic fields
UR - http://eudml.org/doc/210258
ER -

References

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  1. [1] Ph. Cassou-Noguès and M. J. Taylor, Elliptic Functions and Rings of Integers, Progr. Math. 66, Birkhäuser, 1987. Zbl0621.12012
  2. [2] A. Fröhlich, Galois Module Structure of Algebraic Integers, Ergeb. Math. (3) 1, Springer, 1983. Zbl0501.12012
  3. [3] R. Massy, Bases normales d'entiers relatives quadratiques, J. Number Theory 38 (1991), 216-239. Zbl0729.11057
  4. [4] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd ed., Springer, 1990. Zbl0717.11045
  5. [5] K. W. Roggenkamp and M. J. Taylor, Group Rings and Class Groups, DMV Sem. 18, Birkhäuser, 1992. 
  6. [6] M. J. Taylor, Relative Galois module structure of rings of integers, in: Orders and their Applications (Proc. Oberwolfach 1984), I. Reiner and K. W. Roggenkamp (eds.), Lecture Notes in Math. 1142, Springer, 1985, 289-306. 

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