Galois Module Structure of the Integers in Wildly Ramified Cp × Cp Extensions

Elder, G. Griffith ; Madan, Manohar L.

Canadian journal of mathematics, Tome 49 (1997) no. 4, pp. 722-735

Résumé

Let L/K be a finite Galois extension of local fields which are finite extensions of Qp , the field of p-adic numbers. Let Gal(L/K) = G, and OL and Zp be the rings of integers in L and Qp , respectively. And let PL denote the maximal ideal of OL . We determine, explicitly in terms of specific indecomposable Zp [G]-modules, the Zp [G]-module structure of OL and PL , for L, a composite of two arithmetically disjoint, ramified cyclic extensions of K, one of which is only weakly ramified in the sense of Erez [6].

Détail

DOI : 10.4153/CJM-1997-035-2

Mots-clés : 11S15, 20C32, Galois module structure–integral representation

Bibliographie

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

[2] 2. Elder, G.G., The Galois module structure of the integers in wildly ramified extensions, Dissertation, The Ohio State University, 1993. Google Scholar

[3] 3. Elder, G.G. and Madan, M.L., Galois module structure of integers in wildly ramified cyclic extensions. J. Number Theory (2) 47(1994), 138–174. Google Scholar

[4] 4. Elder, G.G., Galois module structure of integers in weakly ramified extensions. Arch. Math. 64(1995), 117–120. Google Scholar

[5] 5. Elder, G.G., Galois module structure of integers in wildly ramified cyclic extensions of degree p2, Ann. Inst. Fourier (Grenoble) (3) 45(1995), 625–647. Google Scholar

[6] 6. Erez, B., The Galois structure of the square root of the inverse different. Math. Z. 208(1991), 239–255. Google Scholar

[7] 7. Fr öhlich, A., Galois Module Structure of Algebraic Integers, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Bd. 1, Springer-Verlag, Berlin, Heidelberg, New York 1983. Google Scholar

[8] 8. Heller, A. and Reiner, I., Representations of cyclic groups in rings of integers I. Ann. of Math. (2) 76(1962), 73–92. Google Scholar

[9] 9. Jaulent, J-F., Sur la l–structure Galoisienne des ideaux ambiges dans une extension metacyclique de degree nl sur le corps des rationnels. Number theory, 1979–1980.and 1980–1981, Exp. 3, 20, Publ. Math. Fac. Sci. Besancon, Univ. Franche–Comté, Besancon, 1981. Google Scholar

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

[11] 11. Miyata, Y., On the module structure of a p-extension over a p-adic number field. Nagoya Math. J. 77(1980), 13–23. Google Scholar

[12] 12. Noether, E., Normalbasis bei Körpern ohne höhere Verzweigung. J. Reine Angew. Math. 167(1932), 147–152. Google Scholar

[13] 13. Rzedowski-Calderón, M., Villa, G.D.-Salvador and Madan, M.L., Galois module structure of rings of integers. Math. Z. 204(1990), 401–424. Google Scholar

[14] 14. Sen, S., On automorphisms of local fields. Ann. of Math. (2) 90(1969), 33–46. Google Scholar

[15] 15. Serre, J-P., Local fields. Graduate Texts Math. 67, Springer-Verlag, Berlin, Heidelberg, New York, 1979. Google Scholar

[16] 16. Shafarevich, I.R., On p-extensions. Izv. Akad. Nauk. SSSR Ser.Mat. 15(1951), 17–46. Google Scholar

[17] 17. Ullom, S., Integral normal bases in Galois extensions of local fields. NagoyaMath. J. 39(1970), 141–146. Google Scholar

[18] 18. Vostokov, S.V., 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. Google Scholar

[19] 19. Yokoi, H., On the ring of integers in an algebraic number field as a representation module of Galois group. Nagoya Math. J. 16(1960), 83–90. Google Scholar