PAC fields over number fields

Moshe Jarden

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Displaying similar documents to “PAC fields over number fields”

On nilpotent Galois groups and the scope of the norm limitation theorem in one-dimensional abstract local class field theory.

Chipchakov, Ivan D. (2005)

Acta Universitatis Apulensis. Mathematics - Informatics

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A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$

Nigel P. Byott (2011)

Journal de Théorie des Nombres de Bordeaux

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Let $S / R$ be a finite extension of discrete valuation rings of characteristic $p > 0$ , and suppose that the corresponding extension $L / K$ of fields of fractions is separable and is $H$ -Galois for some $K$ -Hopf algebra $H$ . Let $𝔻_{S / R}$ be the different of $S / R$ . We show that if $S / R$ is totally ramified and its degree $n$ is a power of $p$ , then any element $ρ$ of $L$ with $v_{L} (ρ) \equiv - v_{L} (𝔻_{S / R}) - 1 (mod n)$ generates $L$ as an $H$ -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois...

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 $ℚ^{(n)} / ℚ^{{(n)}^{+}}$ .

Automatic realizations of Galois groups with cyclic quotient of order $p^{n}$

Ján Mináč, Andrew Schultz, John Swallow (2008)

Journal de Théorie des Nombres de Bordeaux

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We establish automatic realizations of Galois groups among groups $M ⋊ G$ , where $G$ is a cyclic group of order $p^{n}$ for a prime $p$ and $M$ is a quotient of the group ring $𝔽_{p} [G]$ .

Galois Covers and the Hilbert-Grunwald Property

Pierre Dèbes, Nour Ghazi (2012)

Annales de l’institut Fourier

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Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a $p$ -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over $ℚ$ . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.

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 $G ≅ C_{2} \times C_{2}$ 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...

Displaying similar documents to “PAC fields over number fields”

On nilpotent Galois groups and the scope of the norm limitation theorem in one-dimensional abstract local class field theory.

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Note on the Galois module structure of quadratic extensions

Automatic realizations of Galois groups with cyclic quotient of order p n

Galois Covers and the Hilbert-Grunwald Property

Hopf-Galois module structure of tame biquadratic extensions

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$

Automatic realizations of Galois groups with cyclic quotient of order $p^{n}$