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New ramification breaks and additive Galois structure

Nigel P. Byott, G. Griffith Elder (2005)

Journal de Théorie des Nombres de Bordeaux

Which invariants of a Galois p -extension of local number fields L / K (residue field of char p , and Galois group G ) determine the structure of the ideals in L as modules over the group ring ℤ p [ G ] , ℤ p the p -adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups G , we propose and study a new group (within the group ring 𝔽 q [ G ] where 𝔽 q is the residue field) and its resulting ramification filtrations....

Non-existence and splitting theorems for normal integral bases

Cornelius Greither, Henri Johnston (2012)

Annales de l’institut Fourier

We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower ℚ ⊂ K ⊂ L forces the tower to be split in a very strong sense.

Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso, Lorenzo Paolo Rossi (2013)

Acta Arithmetica

We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer extensions...

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, ℚ ( 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 ℚ ( n ) / ℚ ( n ) + .

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