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, . 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 .
@article{bwmeta1.element.bwnjournal-article-cmv67i1p15bwm,
author = {G\"unter Lettl},
title = {Note on the Galois module structure of quadratic extensions},
journal = {Colloquium Mathematicae},
volume = {67},
year = {1994},
pages = {15-19},
zbl = {0812.11063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv67i1p15bwm}
}
Lettl, Günter. Note on the Galois module structure of quadratic extensions. Colloquium Mathematicae, Tome 67 (1994) pp. 15-19. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv67i1p15bwm/
[000] [1] Ph. Cassou-Noguès and M. J. Taylor, Elliptic Functions and Rings of Integers, Progr. Math. 66, Birkhäuser, 1987. | Zbl 0621.12012
[001] [2] A. Fröhlich, Galois Module Structure of Algebraic Integers, Ergeb. Math. (3) 1, Springer, 1983. | Zbl 0501.12012
[002] [3] R. Massy, Bases normales d'entiers relatives quadratiques, J. Number Theory 38 (1991), 216-239. | Zbl 0729.11057
[003] [4] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd ed., Springer, 1990. | Zbl 0717.11045
[004] [5] K. W. Roggenkamp and M. J. Taylor, Group Rings and Class Groups, DMV Sem. 18, Birkhäuser, 1992.
[005] [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.