On Certain Quotients of Grothendieck Groups

1973 ◽  
Vol 16 (3) ◽  
pp. 397-400
Author(s):  
Charles Small
Keyword(s):  
Brauer Group ◽  
Witt Ring ◽  
Grothendieck Groups ◽  
Special Cases ◽  
Characteristic 2

AbstractA general categorical construction is described which has as special cases the construction of the Brauer group of a field and the construction of the Witt ring of a field of characteristic ≠2.

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

On 2-Groups as Galois Groups

1995 ◽  
Vol 47 (6) ◽  
pp. 1253-1273 ◽  
Author(s):  
Arne Ledet
Keyword(s):  
Galois Group ◽  
Direct Product ◽  
Galois Extension ◽  
Quaternion Algebra ◽  
Abelian Groups ◽  
Galois Groups ◽  
Embedding Problem ◽  
Brauer Group ◽  
Characteristic 2 ◽  
Cyclic Kernel

AbstractLet L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

Isomorphisms and Automorphisms of Witt Rings

1988 ◽  
Vol 31 (2) ◽  
pp. 250-256 ◽  
Author(s):  
David Leep ◽  
Murray Marshall
Keyword(s):  
Quadratic Forms ◽  
Multiplicative Group ◽  
Witt Ring ◽  
Witt Rings ◽  
Ring Isomorphism ◽  
Characteristic 2 ◽  
The Relationship

AbstractFor a field F, char(F) ≠ 2, let WF denote the Witt ring of quadratic forms of F and let denote the multiplicative group of 1-dimensional forms It follows from a construction of D. K. Harrison that if E, F are fields (both of characteristic ≠ 2) and ρ.WE → WF is a ring isomorphism, then there exists a ring isomorphism which “preserves dimension” in the sense that In this paper, the relationship between ρ and is clarified.

Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

1993 ◽  
Vol 45 (6) ◽  
pp. 1184-1199 ◽  
Author(s):  
Craig M. Cordes
Keyword(s):  
Group Ring ◽  
Positive Characteristic ◽  
Finitely Generated ◽  
Binary Forms ◽  
Witt Ring ◽  
Witt Rings ◽  
Elementary Type ◽  
Characteristic 2

AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.

Involutory Matrices Over Finite Local Rings

1972 ◽  
Vol 24 (3) ◽  
pp. 369-378 ◽  
Author(s):  
B. R. McDonald
Keyword(s):  
Prime Power ◽  
Quotient Ring ◽  
Careful Analysis ◽  
Commutative Rings ◽  
Local Rings ◽  
Power Characteristic ◽  
Special Cases ◽  
Algebraic Cryptography ◽  
Characteristic 2 ◽  
Finite Commutative Rings

A square matrix A over a commutative ring R is said to be involutory if A2 = I (identity matrix). It has been recognized for some time that involutory matrices have important applications in algebraic cryptography and the special cases where R is either a finite field or a quotient ring of the rational integers have been extensively researched. However, there has been no detailed attempt to extend the theory to all finite commutative rings. In this paper we illustrate in detail the theory of involutory matrices over finite commutative rings with 1 having odd characteristic. The method is a careful analysis of finite local rings of odd prime power characteristic. The techniques might be also used in the examination of involutory matrices over local rings of characteristic 2λ; however, as illustrated by finite fields of characteristic 2 and Z/2λZ (Z the rational integers), the arguments are basically different. The reader will note the methods are not limited to only questions on involutory matrices.

scholarly journals Finiteness of Brauer groups of K3 surfaces in characteristic 2

2018 ◽  
Vol 14 (06) ◽  
pp. 1813-1825
Author(s):  
Kazuhiro Ito
Keyword(s):  
K3 Surfaces ◽  
Torsion Subgroup ◽  
Brauer Group ◽  
Base Field ◽  
Brauer Groups ◽  
Finitely Generated ◽  
Tate Conjecture ◽  
Characteristic 2 ◽  
Primary Torsion

For a [Formula: see text] surface over a field of characteristic [Formula: see text] which is finitely generated over its prime subfield, we prove that the cokernel of the natural map from the Brauer group of the base field to that of the [Formula: see text] surface is finite modulo the [Formula: see text]-primary torsion subgroup. In characteristic different from [Formula: see text], such results were previously proved by Skorobogatov and Zarhin. We basically follow their methods with an extra care in the case of superspecial [Formula: see text] surfaces using the recent results of Kim and Madapusi Pera on the Kuga-Satake construction and the Tate conjecture for [Formula: see text] surfaces in characteristic [Formula: see text].

Electron Beam Damage of Biomolecules Assessed by Energy Loss Spectroscopy

1978 ◽  
Vol 36 (3) ◽  
pp. 61-69
Author(s):  
M. Isaacson ◽  
M.L. Collins ◽  
M. Listvan
Keyword(s):  
Electron Microscopy ◽  
Radiation Damage ◽  
Structural Integrity ◽  
Analytical Electron Microscopy ◽  
High Dose ◽  
Dose Rates ◽  
Special Cases ◽  
Analytical Electron ◽  
Atom Migration ◽  
Beam Damage

Over the past five years it has become evident that radiation damage provides the fundamental limit to the study of blomolecular structure by electron microscopy. In some special cases structural determinations at very low doses can be achieved through superposition techniques to study periodic (Unwin & Henderson, 1975) and nonperiodic (Saxton & Frank, 1977) specimens. In addition, protection methods such as glucose embedding (Unwin & Henderson, 1975) and maintenance of specimen hydration at low temperatures (Taylor & Glaeser, 1976) have also shown promise. Despite these successes, the basic nature of radiation damage in the electron microscope is far from clear. In general we cannot predict exactly how different structures will behave during electron Irradiation at high dose rates. Moreover, with the rapid rise of analytical electron microscopy over the last few years, nvicroscopists are becoming concerned with questions of compositional as well as structural integrity. It is important to measure changes in elemental composition arising from atom migration in or loss from the specimen as a result of electron bombardment.

Decoration technique and surface physics

1978 ◽  
Vol 36 (1) ◽  
pp. 436-437 ◽  
Author(s):  
H. Bethge
Keyword(s):  
Diffusion Path ◽  
Special Importance ◽  
Surface Physics ◽  
Step Structure ◽  
Initial Stage ◽  
Special Cases ◽  
Atomic Surface ◽  
The Mean ◽  
Bulk Structure ◽  
Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

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