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Galois module structure of the rings of integers in wildly ramified extensions

Stephen M. J. Wilson — 1989

Annales de l'institut Fourier

The main results of this paper may be loosely stated as follows. Theorem.— Let N and N ' be sums of Galois algebras with group Γ over algebraic number fields. Suppose that N and N ' have the same dimension ℚ and that they are identical at their wildly ramified primes. Then (writing 𝒪 N for the maximal order in N ) 𝒪 N ⊕ 𝒪 N ⊕ ℤ Γ ≅ ℤ Γ ⊕ 𝒪 N ' ⊕ 𝒪 N ' ⊕ ℤ Γ . In...

Some counter-examples in the theory of the Galois module structure of wild extensions

Stephen M. J. Wilson — 1980

Annales de l'institut Fourier

Considering the ring of integers in a number field as a Z Γ -module (where Γ is a galois group of the field), one hoped to prove useful theorems about the extension of this module to a module or a lattice over a maximal order. In this paper it is show that it could be difficult to obtain, in this way, parameters which are independent of the choice of the maximal order. Several lemmas about twisted group rings are required in the proof.

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