Non-existence and splitting theorems for normal integral bases
[Théorèmes de non-existence et de décomposition pour les bases normales entières]
Greither, Cornelius ; Johnston, Henri
Annales de l'Institut Fourier, Tome 62 (2012), p. 417-437 / Harvested from Numdam
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Nous établissons de nouvelles conditions sous lesquelles il ne peut exister de bases normales entières (faibles) dans les extensions galoisiennes modérées de corps de nombres. Ceci nous conduit au résultat suivant : sous quelques hypothèses techniques convenables, l’existence d’une base normale entière dans l’étage supérieur d’une tour abélienne ℚ⊂K⊂L force que la tour se décompose dans un sens très fort.

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.

Publié le : 2012-01-01
DOI : https://doi.org/10.5802/aif.2709
Classification:  11R33,  11R18,  11R20
Mots clés: base normale entière 
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     author = {Greither, Cornelius and Johnston, Henri},
     title = {Non-existence and splitting theorems for normal integral bases},
     journal = {Annales de l'Institut Fourier},
     volume = {62},
     year = {2012},
     pages = {417-437},
     doi = {10.5802/aif.2709},
     zbl = {1257.11100},
     mrnumber = {2986275},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2012__62_1_417_0}
}

Greither, Cornelius; Johnston, Henri. Non-existence and splitting theorems for normal integral bases. Annales de l'Institut Fourier, Tome 62 (2012) pp. 417-437. doi : 10.5802/aif.2709. http://gdmltest.u-ga.fr/item/AIF_2012__62_1_417_0/

[1] Brinkhuis, J. Normal integral bases and embedding problems, Math. Ann., Tome 264 (1983) no. 4, pp. 537-543 | Article | MR 716266 | Zbl 0516.12005

[2] Brinkhuis, J. Normal integral bases and complex conjugation, J. Reine Angew. Math., Tome 375/376 (1987), pp. 157-166 | MR 882295 | Zbl 0609.12009

[3] Byott, N. P.; Lettl, G. Relative Galois module structure of integers of abelian fields, J. Théor. Nombres Bordeaux, Tome 8 (1996) no. 1, pp. 125-141 | Article | Numdam | MR 1399950 | Zbl 0859.11059

[4] Cougnard, J. Nouveaux exemples d’extension relatives sans base normale, Ann. Fac. Sci. Toulouse Math. (6), Tome 10 (2001) no. 3, pp. 493-505 | Article | Numdam | MR 1923687

[5] Fröhlich, A. Galois module structure of algebraic integers, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 1 (1983) | MR 717033 | Zbl 0501.12012

[6] Fröhlich, A.; Taylor, M. J. Algebraic number theory, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 27 (1993) | MR 1215934 | Zbl 0744.11001

[7] Greither, C. Relative integral normal bases in ℚ(ζ p ), J. Number Theory, Tome 35 (1990) no. 2, pp. 180-193 | Article | MR 1057321 | Zbl 0718.11053

[8] Greither, C. Cyclic Galois extensions of commutative rings, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1534 (1992) | MR 1222646 | Zbl 0788.13003

[9] Lang, S. Cyclotomic fields II, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 69 (1980) | MR 566952 | Zbl 0395.12005

[10] Mcculloh, L. R. Galois module structure of abelian extensions, J. Reine Angew. Math., Tome 375/376 (1987), pp. 259-306 | MR 882300 | Zbl 0619.12008

[11] Washington, L. C. Introduction to cyclotomic fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 83 (1997) | MR 1421575 | Zbl 0484.12001