## Galois module structure of the rings of integers in wildly ramified extensions.(English)Zbl 0674.12005

The main results of this paper may be loosely stated as follows.
Theorem. Let N and $$N'$$ be sums of Galois algebras with group $$\Gamma$$ over algebraic number fields. Suppose that N and $$N'$$ have the same dimension over $${\mathbb{Q}}$$ and that they are identical at their wildly ramified primes. Then (writing $${\mathfrak O}_ N$$ for the maximal order in N) ${\mathfrak O}_ N\oplus {\mathfrak O}_ N\oplus {\mathbb{Z}}\Gamma \cong_{{\mathbb{Z}}\Gamma} {\mathfrak O}_{N'}\oplus {\mathfrak O}_{N'}\oplus {\mathbb{Z}}\Gamma.$ In many cases $${\mathfrak O}_ N \cong_{{\mathbb{Z}}\Gamma} {\mathfrak O}_{N'}.$$
The rôle played by the root numbers of N and $$N'$$ at the symplectic characters of $$\Gamma$$ in determining the relationship between the $${\mathbb{Z}}\Gamma$$-modules $${\mathfrak O}_ N$$ and $${\mathfrak O}_{N'}$$ is described. The theorem includes as a special case the theorem of M. J. Taylor on the structure of the ring of integers in a tamely ramified extension and it employs many of the results employed by Taylor in the proof of his theorem.
Reviewer: S.M.J.Wilson

### MSC:

 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 20C10 Integral representations of finite groups 11R65 Class groups and Picard groups of orders 11R42 Zeta functions and $$L$$-functions of number fields 19A31 $$K_0$$ of group rings and orders
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### References:

 [1] E. BAYER-FLUCKIGER, C. KEARTON and S. M. J. WILSON, Decomposition of modules, forms and simple knots, J. Crelle, 375/376 (1987), 167-183. · Zbl 0607.57013 [2] C. W. CURTIS and I. REINER, Methods of representation theory with applications to finite groups and orders, Wiley, New York, 1983. · Zbl 0616.20001 [3] A. FRÖHLICH, Locally free modules over arithmetic orders, J. Crelle, 274/75 (1975), 112-138. · Zbl 0316.12013 [4] A. FRÖHLICH, Galois module structure of algebraic integers, Springer-Verlag, Berlin, 1983. · Zbl 0501.12012 [5] A. HELLER, Some exact sequences in algebraic K-theory, Topology, 3 (1965), 389-408. · Zbl 0161.01507 [6] J. MARTINET, Character theory and Artin L-functions in “algebraic number fields” (ed. A. Fröhlich), Acad. Press, London, 1977. · Zbl 0359.12015 [7] J. QUEYRUT, Modules radicaux sur des ordres arithmétiques, J. of Algebra, 84 (1983), 420-440. [8] J. QUEYRUT, Anneaux d’entiers dans le même genre, Illinois J. of Math., 29 (1985), 157-179. · Zbl 0552.12005 [9] M. J. TAYLOR, On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math., 63 (1981), 321-353. · Zbl 0469.12003 [10] S. M. J. WILSON, Extensions with identical wild ramification, Séminaire de Théorie des Nombres, Université de Bordeaux I, 1980-1981. · Zbl 0491.12008 [11] S. M. J. WILSON, Structure galoisienne et ramification sauvage, Séminaire de Théorie des Nombres, Université de Bordeaux I, 1986-1987. · Zbl 0682.12004
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