Galois module structure of ideals in wildly ramified cyclic extensions of degree p 2
Annales de l'Institut Fourier, Volume 45 (1995) no. 3, pp. 625-647.

For L/K, any totally ramified cyclic extension of degree p 2 of local fields which are finite extensions of the field of p-adic numbers, we describe the p [ Gal (L/K)]-module structure of each fractional ideal of L explicitly in terms of the 4p+1 indecomposable p [ Gal (L/K)]-modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

Pour n’importe quelle extension cyclique ramifiée de degré p 2 des corps locaux L/K qui sont des extensions finies du corps des nombres p-adiques, nous proposons une description de la p [ Gal (L/K)]-structure de chaque idéal fractionnaire de L utilisant les 4p+1 modules indécomposables sur p [ Gal (L/K)] que Heller et Reiner ont classifié. Les exposants sont entièrement déterminés par les invariants de la ramification.

@article{AIF_1995__45_3_625_0,
     author = {Elder, Gove Griffith},
     title = {Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^2$},
     journal = {Annales de l'Institut Fourier},
     pages = {625--647},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     doi = {10.5802/aif.1468},
     zbl = {0820.11070},
     mrnumber = {96d:11125},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1468/}
}
TY  - JOUR
TI  - Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^2$
JO  - Annales de l'Institut Fourier
PY  - 1995
DA  - 1995///
SP  - 625
EP  - 647
VL  - 45
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1468/
UR  - https://zbmath.org/?q=an%3A0820.11070
UR  - https://www.ams.org/mathscinet-getitem?mr=96d:11125
UR  - https://doi.org/10.5802/aif.1468
DO  - 10.5802/aif.1468
LA  - en
ID  - AIF_1995__45_3_625_0
ER  - 
%0 Journal Article
%T Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^2$
%J Annales de l'Institut Fourier
%D 1995
%P 625-647
%V 45
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1468
%R 10.5802/aif.1468
%G en
%F AIF_1995__45_3_625_0
Elder, Gove Griffith. Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^2$. Annales de l'Institut Fourier, Volume 45 (1995) no. 3, pp. 625-647. doi : 10.5802/aif.1468. https://aif.centre-mersenne.org/articles/10.5802/aif.1468/

[1] A.-M. Bergé, Sur l'arithmétique d'une extension cyclique totalement ramifiée d'un corps local, C. R. Acad. Sc. Paris, 281 (1975), 67-70. | MR | Zbl

[2] F. Bertrandias, Sur les extensions cycliques de degré pn d'un corps local, Acta Arith., 34-4 (1979), 361-377. | MR | Zbl

[3] F. Bertrandias, J.-P. Bertrandias, M.-J. Ferton, Sur l'anneau des entiers d'une extension cyclique de degré premier d'un corps local, C. R. Acad. Sc. Paris, 274 (1972), 1388-1391. | MR | Zbl

[4] N Byott, On Galois isomorphisms between ideals in extensions of local fields, Manuscripta Math., 73 (1991), 289-311. | MR | Zbl

[5] C. W. Curtis, and I. Reiner, Methods of Representation Theory, Wiley, New York, 1981.

[6] G. G. Elder, and M. L. Madan, Galois module structure of integers in wildly ramified cyclic extensions, J. Number Theory, 47 #2 (1994), 138-174. | MR | Zbl

[7] M.-J. Ferton, Sur L'anneau des entiers de certaines extensions cycliques d'un corps local, Astérisque, 24-25 (1975), 21-28. | MR | Zbl

[8] A. Fröhlich, Galois Module Structure of Algebraic Integers, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Bd. 1, Springer-Verlag, Berlin-Heidelberg-New York, 1983. | MR | Zbl

[9] H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der Algebraischen Zahlkörper, Physica-Verlag, Würzburg-Wien, 1970.

[10] H. W. Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math., 201 (1959), 119-149. | EuDML | MR | Zbl

[11] R. E. Mackenzie, and G. Whaples, Artin-Schreier equations in characteristic zero, Am. J. of Math., 78 (1956), 473-485. | MR | Zbl

[12] J. Martinet, Bases normales et constante de l'équation fonctionnelle des fonctions L d'Artin, Séminaire Bourbaki (1973/1974) no. 450. | EuDML | Numdam | Zbl

[13] E. Maus, Existenz β-adischer Zahlkörper zu Vorgegebenem Verzweigungsverhalten, Dissertation, Hamburg, 1965.

[14] H. Miki, On the ramification numbers of cyclic p-extensions over local fields, J. Reine Angew. Math., 328 (1981), 99-115. | EuDML | MR | Zbl

[15] Y. Miyata, On the module structure of a p- extension over a p-adic number field, Nagoya Math. J., 77, (1980), 13-23. | MR | Zbl

[16] M. Rzedowski-Calderón, G. D. Villa-Salvador, M. L. Madan, Galois module structure of rings of integers, Math. Z., 204 (1990), 401-424. | EuDML | MR | Zbl

[17] S. Sen, On automorphisms of local fields, Ann. Math., (2) 90 (1969), 33-46. | MR | Zbl

[18] J-P. Serre, Local fields, Graduate Texts Mathematics, Vol. 67. Springer-Verlag, Berlin-Heidelberg-New York 1979. | Zbl

[19] S. V. Vostokov, Ideals of an abelian p- extension of a local field as Galois modules, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk. SSSR, 57 (1976), 64-84. | Zbl

[20] B. Wyman, Wildly ramified gamma extensions, Am. J. Math., 91 (1969), 135-152. | MR | Zbl

[21] H. Yokoi, On the ring of integers in an algebraic number field as a representation module of Galois group, Nagoya Math. J., 16 (1960), 83-90. | MR | Zbl

Cited by Sources: