no. 1
Non-existence and splitting theorems for normal integral bases
[Théorèmes de non-existence et de décomposition pour les bases normales entières]
Greither, Cornelius1 ; Johnston, Henri2
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, Tome 62 (2012) no. 1, pp. 417-437.

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.

DOI : 10.5802/aif.2709
Classification : 11R33, 11R18, 11R20
Keywords: Normal integral basis
Mot clés : base normale entière
Greither, Cornelius 1 ; Johnston, Henri 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) 
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Greither, Cornelius; Johnston, Henri. Non-existence and splitting theorems for normal integral bases. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 417-437. doi : 10.5802/aif.2709. http://archive.numdam.org/articles/10.5802/aif.2709/

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

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

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

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

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

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

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

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

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

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

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

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