Subfields of henselian valued fields

Ramneek Khassa; Sudesh K. Khanduja

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Displaying similar documents to “Subfields of henselian valued fields”

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $ℚ (α)$ and $ℚ (β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $ℚ (β) \to ℚ (α)$ . The algorithm is particularly efficient if there is only one isomorphism.

Unit vector fields on antipodally punctured spheres: big index, big volume

Fabiano G. B. Brito, Pablo M. Chacón, David L. Johnson (2008)

Bulletin de la Société Mathématique de France

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We establish in this paper a lower bound for the volume of a unit vector field $\vec{v}$ defined on $𝐒^{n} ∖ {\pm x}$ , $n = 2, 3$ . This lower bound is related to the sum of the absolute values of the indices of $\vec{v}$ at $x$ and $- x$ .

$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...

On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Mathematica Bohemica

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We study the capitulation of $2$ -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜 = ℚ (\sqrt{2 p q}, i)$ , where $i = \sqrt{- 1}$ and $p \equiv - q \equiv 1 (mod 4)$ are different primes. For each of the three quadratic extensions $𝕂 / 𝕜$ inside the absolute genus field $𝕜^{(*)}$ of $𝕜$ , we determine a fundamental system of units and then compute the capitulation kernel of $𝕂 / 𝕜$ . The generators of the groups ${Am}_{s} (𝕜 / F)$ and $Am (𝕜 / F)$ are also determined from which we deduce that $𝕜^{(*)}$ is smaller than the relative genus field ${(𝕜 / ℚ (i))}^{*}$ . Then we prove...

The distribution of second $p$ -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

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General concepts and strategies are developed for identifying the isomorphism type of the second $p$ -class group $G = Gal (F_{p}^{2} (K) | K)$ , that is the Galois group of the second Hilbert $p$ -class field $F_{p}^{2} (K)$ , of a number field $K$ , for a prime $p$ . The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢 (p, r)$ , $r \geq 0$ , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$ -class group ${Cl}_{p} (K)$ , the position of $G$ is restricted to certain admissible branches...

Governing fields for the 2-classgroup of $Q (s q r t - q_{1} q_{2} p)$ and a related reciprocity law

Patrick Morton (1990)

Acta Arithmetica

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Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

Lifting vector fields to the rth order frame bundle

J. Kurek, W. M. Mikulski (2008)

Colloquium Mathematicae

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We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^{r} M = i n v J ₀^{r} (ℝ^{m}, M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^{r} M$ . In both cases we deduce that the spaces of all operators in question form free $(m (C_{r}^{m + r} - 1) + 1)$ -dimensional modules over algebras of all smooth maps $J ₀^{r - 1} T ̃ ℝ^{m} \to ℝ$ and $J ₀^{r - 1} T ℝ^{m} \to ℝ$ respectively, where $C ⁿ_{k} = n! / (n - k)! k!$ . We explicitly construct bases of these modules. In particular,...

A $C^{*}$ -Algebra Approach to Field Theory

D. Kastler (1968)

Recherche Coopérative sur Programme n°25

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Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Bertrand Rémy, Amaury Thuillier, Annette Werner (2010)

Annales scientifiques de l'École Normale Supérieure

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We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $G$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $ℬ (G, k)$ to the Berkovich analytic space $G^{an}$ associated with $G$ . Composing this map with the projection of $G^{an}$ to its flag varieties, we define a family of compactifications of $ℬ (G, k)$ . This generalizes results by Berkovich in the case of split groups. Moreover,...

On divisibility of $h^{+}$ by the prime $5$

Stanislav Jakubec (1994)

Mathematica Slovaca

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A remark on the transport equation with b ∈ BV and $d i v_{x} b \in B M O$

Paweł Subko (2014)

Colloquium Mathematicae

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We investigate the transport equation $\partial_{t} u (t, x) + b (t, x) \cdot D_{x} u (t, x) = 0$ . Our result improves the classical criteria of uniqueness of weak solutions in the case of irregular coefficients: b ∈ BV, $d i v_{x} b \in B M O$ . To obtain our result we use a procedure similar to DiPerna and Lions’s one developed for Sobolev vector fields. We apply renormalization theory for BV vector fields and logarithmic type inequalities to obtain energy estimates.

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

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We define a new function space $B$ , which contains in particular BMO, BV, and $W^{1 / p, p}$ , $1 < p < \infty$ . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^{p}$ norms of integer-valued functions in $B$ . We introduce a significant closed subspace, $B_{0}$ , of $B$ , containing in particular VMO and $W^{1 / p, p}$ , $1 \leq p < \infty$ . The above mentioned estimates imply in particular that integer-valued functions belonging to $B_{0}$ are necessarily constant. This framework provides a “common roof”...