Displaying similar documents to “On nilpotent Galois groups and the scope of the norm limitation theorem in one-dimensional abstract local class field theory.”

PAC fields over number fields

Moshe Jarden (2006)

Journal de Théorie des Nombres de Bordeaux

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We prove that if K is a number field and N is a Galois extension of ℚ which is not algebraically closed, then N is not PAC over K .

A classification of the extensions of degree p 2 over ℚ p whose normal closure is a p -extension

Luca Caputo (2007)

Journal de Théorie des Nombres de Bordeaux

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Let k be a finite extension of ℚ p and ℰ k be the set of the extensions of degree p 2 over k whose normal closure is a p -extension. For a fixed discriminant, we show how many extensions there are in ℰ ℚ p with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in ℰ k .

Note on the Galois module structure of quadratic extensions

Günter Lettl (1994)

Colloquium Mathematicae

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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 ) + .

Hopf-Galois module structure of tame biquadratic extensions

Paul J. Truman (2012)

Journal de Théorie des Nombres de Bordeaux

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In [] we studied the nonclassical Hopf-Galois module structure of rings of algebraic integers in some tamely ramified extensions of local and global fields, and proved a partial generalisation of Noether’s theorem to this setting. In this paper we consider tame Galois extensions of number fields L / K with group G ≅ C 2 × C 2 and study in detail the local and global structure of the ring of integers 𝔒 L as a module over its associated order 𝔄 H in each of the Hopf algebras H giving a nonclassical Hopf-Galois...