scholarly journals GEOMETRIC REPRESENTATION IN THE THEORY OF PSEUDO-FINITE FIELDS

2017 ◽  
Vol 82 (3) ◽  
pp. 1132-1139 ◽  
Author(s):  
ÖZLEM BEYARSLAN ◽  
ZOÉ CHATZIDAKIS
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Geometric Representation

AbstractWe study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

scholarly journals The factorable core of polynomials over finite fields

1990 ◽  
Vol 49 (2) ◽  
pp. 309-318 ◽  
Author(s):  
S. D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Permutation Polynomials ◽  
Linearized Polynomial ◽  
Polynomials Over Finite Fields ◽  
Value Sets

AbstractFor a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

Fermat Jacobians of Prime Degree over Finite Fields

1999 ◽  
Vol 42 (1) ◽  
pp. 78-86 ◽  
Author(s):  
Josep González
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Cyclotomic Field ◽  
Roots Of Unity ◽  
Characteristic P ◽  
Prime Degree ◽  
Numerical Criterion

AbstractWe study the splitting of Fermat Jacobians of prime degree l over an algebraic closure of a finite field of characteristic p not equal to l. We prove that their decomposition is determined by the residue degree of p in the cyclotomic field of the l-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

Finite Groups Which are Automorphism Groups of Infinite Groups Only

1985 ◽  
Vol 28 (1) ◽  
pp. 84-90
Author(s):  
Jay Zimmerman
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Schur Multiplier ◽  
Infinite Groups ◽  
Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

2021 ◽  
pp. 2250189
Author(s):  
R. Toledano
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Finite Subset ◽  
Function Fields ◽  
Algebraic Closure ◽  
Value Set ◽  
Towers Of Function Fields ◽  
Value Sets

In this paper, we introduce the notions of [Formula: see text]-polynomial and [Formula: see text]-minimal value set polynomial where [Formula: see text] is a polynomial over a finite field [Formula: see text] and [Formula: see text] is a finite subset of an algebraic closure of [Formula: see text]. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are [Formula: see text]-minimal value set polynomials for [Formula: see text], whose [Formula: see text]-value sets can be explicitly computed in terms of the monomial [Formula: see text].

ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS

2012 ◽  
Vol 08 (04) ◽  
pp. 1087-1097 ◽  
Author(s):  
STEFANIA FANALI ◽  
MASSIMO GIULIETTI
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Direct Product ◽  
Finite Fields ◽  
Classical Problem ◽  
Rational Points ◽  
Fermat Curve ◽  
Cyclic Groups ◽  
Fermat Curves ◽  
Curves Over Finite Fields

The Stöhr–Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves.

Wave Packet Transform over Finite Fields

2015 ◽  
Vol 30 ◽  
pp. 507-529 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Finite Group ◽  
Wave Packet ◽  
Finite Field ◽  
Linear Algebra ◽  
Finite Fields ◽  
Algebra Approach

In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.

scholarly journals Global Norm-Residue Map over Quasi-Finite Field

1966 ◽  
Vol 27 (1) ◽  
pp. 323-329 ◽  
Author(s):  
D. S. Rim ◽  
G. Whaples
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Local Field ◽  
Algebraic Closure ◽  
Residue Field ◽  
Additive Group ◽  
Local Fields ◽  
Ordinary Sense ◽  
Reciprocity Law ◽  
Norm Residue

A field k is called quasi-finite if it is perfect and if Gk≈Ż where Gk is the Galois group of the algebraic closure kc over k and Ż is the completion of the additive group of the rational integers. The classical reciprocity law on the local field with finite residue field is well-known to hold on local fields with quasi-finite residue field ([4] [5]). Thus it is natural to ask if the global reciprocity law should hold in the ordinary sense (see § 1 below) on the function-fields of one variable over quasi-finite field. We consider here two basic prototypes of non-finite quasi-finite fields:

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

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