On matching property for groups and field extensions

In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that ℤ/pℤ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic matching property and show that purely transcendental field extensions satisfy this property.

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Sum of Two Fourth Powers of Integers

Nagoya Mathematical Journal ◽

10.1017/s0027763000002245 ◽

1961 ◽

Vol 18 ◽

pp. 53-61

Author(s):

Minoru Tsunekawa

Keyword(s):

Prime Numbers ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

Problems concerning the sum of two fourth powers of integers seem to be so difficult that little has been known since long years [3]. For instance, it is an important problem to determine whether there are infinitely many prime numbers which are represented in the form p = a4+b4. But nothing is known except that the density of such prime numbers is easily proved to be 0; accordingly it is difficult to obtain a necessary and sufficient condition under which p is represented in such a form.

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Central Extensions of n-Ordered Division Rings

Communications in Algebra ◽

10.1081/agb-200066162 ◽

2005 ◽

Vol 33 (9) ◽

pp. 3265-3281 ◽

Cited By ~ 1

Author(s):

Igor Klep ◽

Dejan Velušček

Keyword(s):

Central Extensions ◽

Division Rings

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FIXED POINTS OF POLYNOMIALS OVER DIVISION RINGS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972721000113 ◽

2021 ◽

pp. 1-7

Author(s):

ADAM CHAPMAN ◽

SOLOMON VISHKAUTSAN

Keyword(s):

Fixed Point ◽

Fixed Points ◽

Periodic Points ◽

Conjugacy Classes ◽

Sufficient Condition ◽

Discrete Dynamics ◽

Polynomial Equations ◽

Commutative Case ◽

Division Rings

Abstract We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

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Central extensions of 𝑆𝐿₂ over division rings and some metaplectic theorems

Contemporary Mathematics - Applications of algebraic K-theory to algebraic geometry and number theory ◽

10.1090/conm/055.2/1862655 ◽

1986 ◽

pp. 561-607 ◽

Cited By ~ 2

Author(s):

Ulf Rehmann

Keyword(s):

Central Extensions ◽

Division Rings

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A deformation problem for Galois representations over imaginary quadratic fields

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748009000036 ◽

2009 ◽

Vol 8 (4) ◽

pp. 669-692 ◽

Cited By ~ 4

Author(s):

Tobias Berger ◽

Krzysztof Klosin

Keyword(s):

Valuation Ring ◽

Quadratic Field ◽

Galois Representations ◽

Discrete Valuation Ring ◽

Quadratic Fields ◽

Sufficient Condition ◽

Imaginary Quadratic Fields ◽

Field Extensions ◽

Deformation Problem ◽

Deformation Ring

AbstractWe prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

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On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

10.5772/intechopen.100021 ◽

2021 ◽

Author(s):

Lhoussain El Fadil ◽

Mohamed Faris

Keyword(s):

Algebraic Number Theory ◽

Sufficient Condition ◽

Polynomial Factorization ◽

General Version ◽

Field Extensions ◽

Valued Field ◽

Rank One ◽

Key Polynomials ◽

Irreducibility Criterion

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L=Kα, we give a method to describe all absolute values of L extending ∣∣, where K is a discrete rank one valued field.

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The Current Situation in Chemical Stabilization of Biological Materials

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100093912 ◽

1972 ◽

Vol 30 ◽

pp. 132-133

Author(s):

John H. Luft

Keyword(s):

Electron Microscopy ◽

Electron Microscope ◽

Osmium Tetroxide ◽

Molecular Level ◽

Cross Linking ◽

Chemical Stabilization ◽

Specimen Preparation ◽

Sufficient Condition ◽

Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

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