Non-existence and splitting theorems for normal integral bases

Cornelius Greither[1]; Henri Johnston[2]

  • [1] Universität der Bundeswehr München Fakultät für Informatik Institut für theoretische Informatik und Mathematik 85577 Neubiberg (Germany)
  • [2] St. John’s College Cambridge CB2 1TP (United Kingdom)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 1, page 417-437
  • ISSN: 0373-0956

Abstract

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We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower ℚ ⊂ K ⊂ L forces the tower to be split in a very strong sense.

How to cite

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Greither, Cornelius, and Johnston, Henri. "Non-existence and splitting theorems for normal integral bases." Annales de l’institut Fourier 62.1 (2012): 417-437. <http://eudml.org/doc/251127>.

@article{Greither2012,
abstract = {We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower $\mathbb\{Q\} \subset K \subset L$ forces the tower to be split in a very strong sense.},
affiliation = {Universität der Bundeswehr München Fakultät für Informatik Institut für theoretische Informatik und Mathematik 85577 Neubiberg (Germany); St. John’s College Cambridge CB2 1TP (United Kingdom)},
author = {Greither, Cornelius, Johnston, Henri},
journal = {Annales de l’institut Fourier},
keywords = {Normal integral basis; (weak) normal integral basis; Hilbert-Speiser Theorem; arithmetically disjoint; arithmetically split; Amitsur cohomology},
language = {eng},
number = {1},
pages = {417-437},
publisher = {Association des Annales de l’institut Fourier},
title = {Non-existence and splitting theorems for normal integral bases},
url = {http://eudml.org/doc/251127},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Greither, Cornelius
AU - Johnston, Henri
TI - Non-existence and splitting theorems for normal integral bases
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 1
SP - 417
EP - 437
AB - We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower $\mathbb{Q} \subset K \subset L$ forces the tower to be split in a very strong sense.
LA - eng
KW - Normal integral basis; (weak) normal integral basis; Hilbert-Speiser Theorem; arithmetically disjoint; arithmetically split; Amitsur cohomology
UR - http://eudml.org/doc/251127
ER -

References

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  1. J. Brinkhuis, Normal integral bases and embedding problems, Math. Ann. 264 (1983), 537-543 Zbl0516.12005MR716266
  2. J. Brinkhuis, Normal integral bases and complex conjugation, J. Reine Angew. Math. 375/376 (1987), 157-166 Zbl0609.12009MR882295
  3. N. P. Byott, G. Lettl, Relative Galois module structure of integers of abelian fields, J. Théor. Nombres Bordeaux 8 (1996), 125-141 Zbl0859.11059MR1399950
  4. J. Cougnard, Nouveaux exemples d’extension relatives sans base normale, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 493-505 MR1923687
  5. A. Fröhlich, Galois module structure of algebraic integers, 1 (1983), Springer-Verlag, Berlin Zbl0501.12012MR717033
  6. A. Fröhlich, M. J. Taylor, Algebraic number theory, 27 (1993), Cambridge University Press, Cambridge Zbl0744.11001MR1215934
  7. C. Greither, Relative integral normal bases in ℚ ( ζ p ) , J. Number Theory 35 (1990), 180-193 Zbl0718.11053MR1057321
  8. C. Greither, Cyclic Galois extensions of commutative rings, 1534 (1992), Springer-Verlag, Berlin Zbl0788.13003MR1222646
  9. S. Lang, Cyclotomic fields II, 69 (1980), Springer-Verlag, New York Zbl0395.12005MR566952
  10. L. R. McCulloh, Galois module structure of abelian extensions, J. Reine Angew. Math. 375/376 (1987), 259-306 Zbl0619.12008MR882300
  11. L. C. Washington, Introduction to cyclotomic fields, 83 (1997), Springer-Verlag, New York Zbl0484.12001MR1421575

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