Galois module structure of ideals in wildly ramified cyclic extensions of degree p 2
Elder, Gove Griffith
Annales de l'Institut Fourier, Tome 45 (1995), p. 625-647 / Harvested from Numdam

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.

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.

@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},
     volume = {45},
     year = {1995},
     pages = {625-647},
     doi = {10.5802/aif.1468},
     mrnumber = {96d:11125},
     zbl = {0820.11070},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 45 (1995) pp. 625-647. doi : 10.5802/aif.1468. http://gdmltest.u-ga.fr/item/AIF_1995__45_3_625_0/

[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 54 #2625 | Zbl 0306.12009

[2] F. Bertrandias, Sur les extensions cycliques de degré pn d'un corps local, Acta Arith., 34-4 (1979), 361-377. | MR 80k:12022 | Zbl 0381.12008

[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 45 #5109 | Zbl 0235.12008

[4] N Byott, On Galois isomorphisms between ideals in extensions of local fields, Manuscripta Math., 73 (1991), 289-311. | MR 92g:11115 | Zbl 0771.11047

[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 95e:11125 | Zbl 0801.11046

[7] M.-J. Ferton, Sur L'anneau des entiers de certaines extensions cycliques d'un corps local, Astérisque, 24-25 (1975), 21-28. | MR 51 #10305 | Zbl 0306.12008

[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 85h:11067 | Zbl 0501.12012

[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. | MR 21 #7195 | Zbl 0098.03403

[11] R. E. Mackenzie, and G. Whaples, Artin-Schreier equations in characteristic zero, Am. J. of Math., 78 (1956), 473-485. | MR 19,834c | Zbl 0073.26402

[12] J. Martinet, Bases normales et constante de l'équation fonctionnelle des fonctions L d'Artin, Séminaire Bourbaki (1973/1974) no. 450. | Numdam | Zbl 0331.12006

[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. | MR 83k:12014 | Zbl 0457.12005

[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 81b:12015 | Zbl 0444.12012

[16] M. Rzedowski-Calderón, G. D. Villa-Salvador, M. L. Madan, Galois module structure of rings of integers, Math. Z., 204 (1990), 401-424. | MR 92e:11128 | Zbl 0682.12003

[17] S. Sen, On automorphisms of local fields, Ann. Math., (2) 90 (1969), 33-46. | MR 39 #5531 | Zbl 0199.36301

[18] J-P. Serre, Local fields, Graduate Texts Mathematics, Vol. 67. Springer-Verlag, Berlin-Heidelberg-New York 1979. | Zbl 0423.12016

[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 0355.12012

[20] B. Wyman, Wildly ramified gamma extensions, Am. J. Math., 91 (1969), 135-152. | MR 39 #2726 | Zbl 0188.11003

[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 23 #A888 | Zbl 0119.27703