Computing p th roots in extended finite fields of prime characteristic p ≥ 2

2016 ◽  
Vol 52 (9) ◽  
pp. 718-719
Author(s):  
M. Repka
Keyword(s):  
Finite Fields ◽  
Prime Characteristic ◽  
Characteristic P

scholarly journals m-NIL-CLEAN COMPANION MATRICES

2019 ◽  
Vol 35 ◽  
pp. 626-632
Author(s):  
Andrada Cîmpean
Keyword(s):  
Positive Characteristic ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Companion Matrices

Companion matrices over fields of prime characteristic, p, that are sums of two idempotents and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p. Companion matrices over fields of positive characteristic, p, that are sums of m idempotents, m ≥ 2, and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p.

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.

The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

2017 ◽  
Vol 24 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Shuangnian Hu ◽  
Junyong Zhao
Keyword(s):  
Number Theory ◽  
Normal Form ◽  
Finite Fields ◽  
Rational Points ◽  
System Of Equations ◽  
Algebraic Varieties ◽  
Natural Conditions ◽  
Characteristic P ◽  
Exponent Matrix ◽  
Positive Integers

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

Certain 3-decompositions of complete graphs, with an application to finite fields

1985 ◽  
Vol 99 (3-4) ◽  
pp. 277-281 ◽  
Author(s):  
Peter Rowlinson
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Complete Graph ◽  
Disjoint Union ◽  
Necessary Condition ◽  
Complete Graphs ◽  
Characteristic P ◽  
Strongly Regular

SynopsisA necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.

scholarly journals Commutative rings with homomorphic power functions

1992 ◽  
Vol 15 (1) ◽  
pp. 91-102
Author(s):  
David E. Dobbs ◽  
John O. Kiltinen ◽  
Bobby J. Orndorff
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Structure Theorem ◽  
Commutative Rings ◽  
Prime Characteristic ◽  
Power Functions ◽  
Finite Case

A (commutative) ringR(with identity) is calledm-linear (for an integerm≥2) if(a+b)m=am+bmfor allaandbinR. Them-linear reduced rings are characterized, with special attention to the finite case. A structure theorem reduces the study ofm-linearity to the case of prime characteristic, for which the following result establishes an analogy with finite fields. For each primepand integerm≥2which is not a power ofp, there exists an integers≥msuch that, for each ringRof characteristicp,Rism-linear if and only ifrm=rpsfor eachrinR. Additional results and examples are given.

On Unit Groups in Modular Group Algebras of Abelian Groups with Totally Projective Components

2007 ◽  
Vol 14 (03) ◽  
pp. 515-520
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Modular Group ◽  
Abelian Groups ◽  
Group Algebras ◽  
Prime Characteristic ◽  
Characteristic P

We prove that if the p-reduced abelian group G is a special countable extension of its totally projective p-component of torsion Gp and R is a perfect commutative unitary ring of prime characteristic p, then the group S(G) of all normed p-units in the group algebra RG modulo Gp, that is, S(G)/Gp, is totally projective. Our result strengthens both classical results obtained by May and Hill–Ullery.

On Katz’s (A,B)-exponential sums

2020 ◽  
Author(s):  
Lei FU ◽  
Daqing WAN
Keyword(s):  
Finite Fields ◽  
Arithmetic Progression ◽  
Betti Number ◽  
Exponential Sums ◽  
Decomposition Theorem ◽  
Characteristic P ◽  
Universal Family ◽  
Degenerate Cases

Abstract We deduce Katz’s theorems for (A, B)-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef–Loeser, removing the hypothesis that A + B is relatively prime to the characteristic p. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson–Sperber’s bounds for degrees of L-functions. Applying the facial decomposition theorem, we prove that the universal family of (A, B)-polynomials is generically ordinary for its L-function when p is in certain arithmetic progression.

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