Factorisability and wildly ramified Galois extensions
Annales de l'Institut Fourier, Tome 41 (1991) no. 2, pp. 393-430.

Soit L/K une extension abélienne de corps p-adiques, et soit 𝒪 l’anneau de valuation de K. Nous étudions les idéaux de l’anneau de valuation de L comme représentations entières du groupe de Galois Gal (L/K). À supposer que K soit absolument non ramifiée nous utilisons les techniques de la théorie de la factorisabilité pour examiner quels idéaux sont isomorphes à un 𝒪-ordre dans l’algèbre du groupe K[ Gal (l/K)]. Nous obtenons de nouveaux résultats généraux et aussi explicites.

Let L/K be an abelian extension of p-adic fields, and let 𝒪 denote the valuation ring of K. We study ideals of the valuation ring of L as integral representations of the Galois group Gal (L/K). Assuming K is absolutely unramified we use techniques from the theory of factorisability to investigate which ideals are isomorphic to an 𝒪-order in the group algebra K[ Gal (l/K)]. We obtain several general and also explicit new results.

@article{AIF_1991__41_2_393_0,
     author = {Burns, David J.},
     title = {Factorisability and wildly ramified {Galois} extensions},
     journal = {Annales de l'Institut Fourier},
     pages = {393--430},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {2},
     year = {1991},
     doi = {10.5802/aif.1259},
     mrnumber = {92m:11135},
     zbl = {0727.11048},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1259/}
}
TY  - JOUR
AU  - Burns, David J.
TI  - Factorisability and wildly ramified Galois extensions
JO  - Annales de l'Institut Fourier
PY  - 1991
SP  - 393
EP  - 430
VL  - 41
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1259/
DO  - 10.5802/aif.1259
LA  - en
ID  - AIF_1991__41_2_393_0
ER  - 
%0 Journal Article
%A Burns, David J.
%T Factorisability and wildly ramified Galois extensions
%J Annales de l'Institut Fourier
%D 1991
%P 393-430
%V 41
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1259/
%R 10.5802/aif.1259
%G en
%F AIF_1991__41_2_393_0
Burns, David J. Factorisability and wildly ramified Galois extensions. Annales de l'Institut Fourier, Tome 41 (1991) no. 2, pp. 393-430. doi : 10.5802/aif.1259. http://www.numdam.org/articles/10.5802/aif.1259/

[1] A.-M. Bergé, Arithmétique d'une extension à groupe d'inertie cyclique, Ann. Inst. Fourier, 28, 4 (1978), 17-44. | Numdam | MR | Zbl

[2] A.-M. Bergé, A propos du genre de l'anneau des entiers d'une extension, Publications Math. Sc. Besançon, (1979-1980), 1-9. | MR | Zbl

[3] D. Burns, Factorisability, group lattices and Galois module structure, J. of Algebra, 134 (1990), 257-270. | Zbl

[4] D. Burns, Canonical factorisability and a variant of Martinet's conjecture, to appear in J. London Math. Soc., (1991). | MR | Zbl

[5] S. U. Chase and F. Destrempes, Factorizability, Grothendieck groups and Galois module structure, preprint, 1989.

[6] J. W. S. Cassels and A. Fröhlich (eds), Algebraic Number Theory, Proc. Brighton Symp., 1965, Academic Press, London, 1967. | Zbl

[7] Ph. Cassou-Noguès and M. J. Taylor, Elliptic functions and rings of integers, Progress in Mathematics, Volume 66, Birkhäuser Boston-Basel-Stuttgart, 1987. | MR | Zbl

[8] B. Erez, A survey of recent work on the square root of the inverse different, Proceedings of the Journées Arithmétique 1989 at Luminy. | Numdam | Zbl

[9] M.-J. Ferton, Sur les idéaux d'une extension cyclique de degré premier d'un corps local, C.R. Acad. Sc. Paris, 276 Série A (1973), 1483-1486. | MR | Zbl

[10] J.-M. Fontaine, Groupes de ramification et représentations d'Artin, Ann. Scient. Éc. Norm. Sup., 4e série, 4 (1971), 337-392. | Numdam | MR | Zbl

[11] A. Fröhlich, Invariants for modules over commutative separable orders, Quart. J. Math. Oxford, 16 (1965), 193-232. | MR | Zbl

[12] A. Fröhlich, Module defect and factorisability, Illinois J. Math., 32, 3 (1988), 407-421. | MR | Zbl

[13] A. Fröhlich, L-values at zero and multiplicative Galois module structure (also Galois Gauss sums and additive Galois module structure), J. reine und angew. Math., 397 (1989), 42-99. | MR | Zbl

[14] F. Kawamoto, On normal integral bases of local fields, J. of Algebra, 98 (1986), 197-199. | MR | Zbl

[15] H. W. Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine und angew. Math., 201 (1959), 119-149. | MR | Zbl

[16] A. Nelson, Monomial representations and Galois module structure, Ph. D. Thesis, King's College, University of London, 1979.

[17] I. Reiner, Maximal Orders, Academic Press, London, 1975. | MR | Zbl

[18] J.-P. Serre, Corps Locaux, Hermann, Paris, 1962. | MR | Zbl

[19] D. Solomon, Iwasawa theory, factorisability and the Galois module structure of units, to appear. | Zbl

[20] R. G. Swan, Induced representations and projective modules, Ann. of Math., 71 (1960), 552-578. | MR | Zbl

[21] S. Ullom, Normal bases in Galois extensions of number fields, Nagoya J., 34 (1969), 153-167. | MR | Zbl

[22] S. Ullom, Galois cohomology of Ambiguous Ideals, J. Number Theory, 1 (1969), 11-15. | MR | Zbl

[23] S. M. J. Wilson, Extensions with identical wild ramification, Sém. de Théorie des Nombres, Université de Bordeaux I, (1980-1981). | Zbl

Cité par Sources :