# Note on the Galois module structure of quadratic extensions

Colloquium Mathematicae (1994)

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

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topLettl, 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

top- [1] Ph. Cassou-Noguès and M. J. Taylor, Elliptic Functions and Rings of Integers, Progr. Math. 66, Birkhäuser, 1987. Zbl0621.12012
- [2] A. Fröhlich, Galois Module Structure of Algebraic Integers, Ergeb. Math. (3) 1, Springer, 1983. Zbl0501.12012
- [3] R. Massy, Bases normales d'entiers relatives quadratiques, J. Number Theory 38 (1991), 216-239. Zbl0729.11057
- [4] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd ed., Springer, 1990. Zbl0717.11045
- [5] K. W. Roggenkamp and M. J. Taylor, Group Rings and Class Groups, DMV Sem. 18, Birkhäuser, 1992.
- [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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