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Note on the Galois module structure of quadratic extensions

Günter Lettl — 1994

Colloquium Mathematicae

In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, ( p ) . For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of ( n ) / ( n ) + .

Relative Galois module structure of integers of abelian fields

Nigel P. ByottGünter Lettl — 1996

Journal de théorie des nombres de Bordeaux

Let L / K be an extension of algebraic number fields, where L is abelian over . In this paper we give an explicit description of the associated order 𝒜 L / K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L , is then isomorphic to 𝒜 L / K . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that 𝒜 L / K is the maximal order if L / K is a cyclic and totally wildly ramified extension which is linearly disjoint to ( m ' ) / K , where m ' is the conductor of K .

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