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


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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