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

Published online by Cambridge University Press:  20 November 2018

G. Griffith Elder  and
Manohar L. Madan
G. Griffith Elder
Affiliation:
The Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210, U.S.A.
Manohar L. Madan
Affiliation:
The Department of Mathematics, The University of Nebraska at Omaha, Omaha, Nebraska 68182, U.S.A.
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Abstract

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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].

Keywords

11S1520C32Galois module structure–integral representation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 4 , 01 August 1997 , pp. 722 - 735
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Curtis, C.W. and Reiner, I., Methods of Representation Theory, Wiley, New York, 1981.Google Scholar
2. Elder, G.G., The Galois module structure of the integers in wildly ramified extensions, Dissertation, The Ohio State University, 1993.Google Scholar
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. Elder, G.G., Galois module structure of integers in weakly ramified extensions. Arch. Math. 64(1995), 117–120.Google Scholar
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. Erez, B., The Galois structure of the square root of the inverse different. Math. Z. 208(1991), 239–255.Google Scholar
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. 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. 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. Maus, E., Existenz β-adischer Zahlkörper zu Vorgegebenem Verzweigungsverhalten, Dissertation, Hamburg, 1965.Google Scholar
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. Noether, E., Normalbasis bei Körpern ohne höhere Verzweigung. J. Reine Angew. Math. 167(1932), 147–152.Google Scholar
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. Sen, S., On automorphisms of local fields. Ann. of Math. (2) 90(1969), 33–46.Google Scholar
15. Serre, J-P., Local fields. Graduate Texts Math. 67, Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
16. Shafarevich, I.R., On p-extensions. Izv. Akad. Nauk. SSSR Ser.Mat. 15(1951), 17–46.Google Scholar
17. Ullom, S., Integral normal bases in Galois extensions of local fields. NagoyaMath. J. 39(1970), 141–146.Google Scholar
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. 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