scholarly journals Quadratic subfields on quartic extensions of local fields

1988 ◽  
Vol 11 (1) ◽  
pp. 1-4
Author(s):  
Joe Repka
Keyword(s):  
Galois Group ◽  
Local Field ◽  
Local Fields ◽  
Intermediate Field ◽  
Residue Characteristic

We show that any quartic extension of a local field of odd residue characteristic must contain an intermediate field. A consequence of this is that local fields of odd residue characteristic do not have extensions with Galois groupA4orS4. Counterexamples are given for even residue characteristic.

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

Local Field Effects Near Surfaces of Small Systems

1994 ◽  
Vol 366 ◽  
Author(s):  
David Beaglehole
Keyword(s):  
Dielectric Constant ◽  
External Field ◽  
Electric Field ◽  
Local Field ◽  
Computer Calculation ◽  
Local Fields ◽  
Small Systems ◽  
Shape Dependence ◽  
Self Consistent ◽  
Reflection Of Light

ABSTRACTThe interaction of light with a system of molecules depends upon the polarisation induced by an external electric field, which depends not only upon the external field but also upon the local fields due to neighboring polarised molecules. These local fields result in the traditional Clausius-Mossotti (CM) dielectric constant for a molecule deeply imbedded in a medium. Near the surface the local fields are altered, and the dielectric constant becomes anisotropic and dependent upon depth into the medium. The local fields are shape dependent in small systems and differ substantially from the CM value.A self-consistent computer calculation of the local fields has been implemented, and these effects will be shown using molecule positions and polarisabilities typical of liquids and crystals. The shape dependence of small systems, the reflection of light from liquids with fluctuating surfaces, and the effect of supporting substrates will be described.

scholarly journals CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

2018 ◽  
Vol 19 (4) ◽  
pp. 1031-1091
Author(s):  
Thierry Stulemeijer
Keyword(s):  
Algebraic Group ◽  
Closed Subset ◽  
Algebraic Groups ◽  
Local Fields ◽  
Locally Finite ◽  
Simple Algebraic Group ◽  
Groups Acting On Trees ◽  
Residue Characteristic

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

scholarly journals Fundamental theorem of prehomogeneous vector spaces of characteristic p

1997 ◽  
Vol 56 (2) ◽  
pp. 331-341
Author(s):  
Tatsuo Kimura ◽  
Takeyoshi Kogiso ◽  
Makiko Fujinaga
Keyword(s):  
Local Field ◽  
Fundamental Theorem ◽  
Functional Equations ◽  
Local Fields ◽  
Vector Spaces ◽  
Characteristic P ◽  
Prehomogeneous Vector Spaces

For a local field of characteristic 0, the functional equations of zeta distributions of prehomogeneous vector spaces have been obtained by M. Sato, Shintani, Igusa, F. Sato and Gyoja. In this paper, we shall consider the case of local fields of characteristic p > 0.

scholarly journals CONSTRUCTION OF SELF-DUAL INTEGRAL NORMAL BASES IN ABELIAN EXTENSIONS OF FINITE AND LOCAL FIELDS

2010 ◽  
Vol 06 (07) ◽  
pp. 1565-1588 ◽  
Author(s):  
ERIK JARL PICKETT
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Galois Extension ◽  
Valuation Ring ◽  
Finite Extension ◽  
Normal Basis ◽  
Local Fields ◽  
Abelian Extensions ◽  
Construction Of Self ◽  
Odd Degree

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that Tr F/E(g(x), h(x)) = δg, h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char (E) ≠ 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char (E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of ℚp, let L/K be a finite abelian Galois extension of odd degree and let [Formula: see text] be the valuation ring of L. We define AL/K to be the unique fractional [Formula: see text]-ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for AL/K if and only if L/K is weakly ramified. Assuming p ≠ 2, we construct such bases whenever they exist.

scholarly journals Dualwavelet frames in Sobolev spaces on local fields of positive characteristic

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 2091-2099
Author(s):  
Ishtaq Ahmad ◽  
Neyaz Sheikh
Keyword(s):  
Sobolev Spaces ◽  
Local Field ◽  
Positive Characteristic ◽  
Local Fields ◽  
Wavelet Frames ◽  
The Past ◽  
Dual Wavelet Frames ◽  
Dual Wavelet ◽  
Field Of Positive Characteristic

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. In this article, we obtain the characterization of nonhomogeneous wavelet frames and nonhomogeneous dual wavelet frames in a Sobolev spaces on a local field of positive characteristic by means of a pair of equations.

scholarly journals The Galois group of local fields

1977 ◽  
Vol 29 (3) ◽  
pp. 385-416 ◽  
Author(s):  
Sin-Ichi Watanabe
Keyword(s):  
Galois Group ◽  
Local Fields
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