scholarly journals On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

2021 ◽  
Vol 14 (5) ◽  
pp. 547-553
Author(s):  
Peter Danchev ◽  
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Closed Field ◽  
Multilinear Algebra ◽  
Infinite Field ◽  
Algebraically Closed Field ◽  
Periodic Matrix ◽  
Nilpotent Matrix ◽  
Square Matrix

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

scholarly journals Abelian varieties over finite fields as basic abelian varieties

2017 ◽  
Vol 29 (2) ◽  
pp. 489-500 ◽  
Author(s):  
Chia-Fu Yu
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Finite Fields ◽  
Mass Formula ◽  
Abelian Varieties ◽  
Closed Field ◽  
Abelian Surfaces ◽  
Algebraically Closed Field

AbstractIn this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic ${p>0}$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

scholarly journals Certain Properties of Square Matrices over Fields with Applications to Rings

2021 ◽  
Vol 54 (2) ◽  
pp. 109-116
Author(s):  
Peter V. Danchev
Keyword(s):  
Linear Algebra ◽  
Present Author ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Matrix ◽  
Zero Matrix ◽  
Siberian Math ◽  
Diagonalizable Matrix ◽  
Regular Rings ◽  
Square Matrix

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.

Matrix Commutators

1961 ◽  
Vol 13 ◽  
pp. 353-355 ◽  
Author(s):  
M. F. Smiley
Keyword(s):  
Closed Field ◽  
Classical Theorem ◽  
Algebraically Closed Field ◽  
Square Matrix

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

2021 ◽  
Vol 21 (4) ◽  
pp. 451-461
Author(s):  
Massimo Giulietti ◽  
Motoko Kawakita ◽  
Stefano Lia ◽  
Maria Montanucci
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Finite Field ◽  
Symmetric Group ◽  
Closed Field ◽  
Complex Field ◽  
Hermitian Curve ◽  
Full Automorphism Group ◽  
Algebraically Closed Field

Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)−12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

scholarly journals ASSOCIATIVE NIL-ALGEBRAS OVER FINITE FIELDS

2013 ◽  
Vol 23 (08) ◽  
pp. 1881-1894 ◽  
Author(s):  
ARTEM A. LOPATIN ◽  
IVAN P. SHESTAKOV
Keyword(s):  
Finite Field ◽  
Associative Algebra ◽  
Finite Fields ◽  
Infinite Field ◽  
Finitely Generated

We study the nilpotency degree of a relatively free finitely generated associative algebra with the identity xn = 0 over a finite field 𝔽 with q elements. In the case of q ≥ n the nilpotency degree is proven to be the same as in the case of an infinite field of the same characteristic. In the case of q = n - 1 it is shown that the nilpotency degree differs from the nilpotency degree for an infinite field of the same characteristic by at most one. The nilpotency degree is explicitly computed for n = 3.

Rings which admit elimination of quantifiers

1978 ◽  
Vol 43 (1) ◽  
pp. 92-112 ◽  
Author(s):  
Bruce I. Rose
Keyword(s):  
Finite Field ◽  
Prime Ring ◽  
Chain Condition ◽  
Closed Field ◽  
Matrix Rings ◽  
Algebraically Closed Field ◽  
Finite Set ◽  
Descending Chain Condition ◽  
Infinite Center ◽  
Ordered Pairs

AbstractWe say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers.Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field.A ring is prime if it satisfies the sentence: ∀x∀y∃z (x =0 ∨ y = 0∨ xzy ≠ 0).Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn)⊕GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let be the set of ordered pairs (f, Q) where Q is a finite set of primes and such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q ∈ Qf(q), for some (f, Q) in .Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to.Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to.In contrast to Theorems 2 and 4, we haveTheorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition.We also generalize Theorems 1, 2 and 4 to alternative rings.

scholarly journals Polarizations on abelian varieties

2002 ◽  
Vol 133 (2) ◽  
pp. 223-233 ◽  
Author(s):  
A. SILVERBERG ◽  
YU. G. ZARHIN
Keyword(s):  
Abelian Variety ◽  
Finite Fields ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Number Fields ◽  
General Context ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic

Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]2. We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]2.

On splitting of the normalizer of a maximal torus in linear groups

2015 ◽  
Vol 14 (07) ◽  
pp. 1550114 ◽  
Author(s):  
Alexey Galt
Keyword(s):  
Finite Field ◽  
Maximal Torus ◽  
Linear Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Maximal Tori

We describe linear groups over an algebraically closed field in which the normalizer of a maximal torus splits over the torus. We describe linear groups over a finite field and their maximal tori in which the normalizer of the maximal torus splits over the torus.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

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