The Experts below are selected from a list of 2766 Experts worldwide ranked by ideXlab platform
Stephen D. Cohen - One of the best experts on this subject based on the ideXlab platform.
-
Primitive Element Pairs with a Prescribed Trace in the Quartic Extension of a Finite Field.
Journal of Algebra and Its Applications, 2020Co-Authors: Stephen D. Cohen, Anju GuptaAbstract:In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a Primitive Element $\alpha $ such that the Element $\alpha+\alpha^{-1}$ is also a Primitive Element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.
-
Existence results for Primitive Elements in cubic and quartic extensions of a finite field
Mathematics of Computation, 2018Co-Authors: Geoff Bailey, Stephen D. Cohen, Nicole Sutherland, Tim TrudgianAbstract:With $\Fq$ the finite field of $q$ Elements, we investigate the following question. If $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a non-zero Element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$ is a Primitive Element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$.
-
Primitive Element pairs with one prescribed trace over a finite field
Finite Fields and Their Applications, 2018Co-Authors: Anju Gupta, Rajendra K. Sharma, Stephen D. CohenAbstract:Abstract In this article, we establish a sufficient condition for the existence of a Primitive Element α ∈ F q n such that the Element α + α − 1 is also a Primitive Element of F q n , and T r F q n | F q ( α ) = a for any prescribed a ∈ F q , where q = p k for some prime p and positive integer k. We prove that every finite field F q n ( n ≥ 5 ) , contains such Primitive Elements except for finitely many values of q and n. Indeed, by computation, we conclude that there are no actual exceptional pairs ( q , n ) for n ≥ 5 .
-
Primitive Element Pairs with One Prescribed Trace over a Finite Field
arXiv: Number Theory, 2017Co-Authors: Anju Gupta, Rajendra K. Sharma, Stephen D. CohenAbstract:In this article, we establish a sufficient condition for the existence of a Primitive Element $\alpha \in {\mathbb{F}_{q^n}}$ such that the Element $\alpha+\alpha^{-1}$ is also a Primitive Element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains such Primitive Elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\geq5.$
-
Pairs of Primitive Elements in fields of even order
Finite Fields and Their Applications, 2014Co-Authors: Stephen D. CohenAbstract:Let F q be a finite field of even order. Two existence theorems, towards which partial results have been obtained by Wang, Cao and Feng, are now established. These state that (i) for any q ? 8 , there exists a Primitive Element α ? F q such that α + 1 / α is also Primitive, and (ii) for any integer n ? 3 , in the extension of degree n over F q there exists a Primitive Element α with α + 1 / α also Primitive such that α is a normal Element over F q .Corresponding results for finite fields of odd order remain to be investigated.
Rajendra K. Sharma - One of the best experts on this subject based on the ideXlab platform.
-
Pair of Primitive Elements with prescribed traces over finite fields
Communications in Algebra, 2019Co-Authors: Rajendra K. Sharma, Anju GuptaAbstract:In this article, we establish a sufficient condition for the existence of a Primitive Element α∈Fqn such that for any matrix (abc0de)∈M2×3(Fqn) of rank 2, the Element (aα2+bα+c)/(dα+e) is a primiti...
-
Primitive Element pairs with one prescribed trace over a finite field
Finite Fields and Their Applications, 2018Co-Authors: Anju Gupta, Rajendra K. Sharma, Stephen D. CohenAbstract:Abstract In this article, we establish a sufficient condition for the existence of a Primitive Element α ∈ F q n such that the Element α + α − 1 is also a Primitive Element of F q n , and T r F q n | F q ( α ) = a for any prescribed a ∈ F q , where q = p k for some prime p and positive integer k. We prove that every finite field F q n ( n ≥ 5 ) , contains such Primitive Elements except for finitely many values of q and n. Indeed, by computation, we conclude that there are no actual exceptional pairs ( q , n ) for n ≥ 5 .
-
Primitive Element Pairs with One Prescribed Trace over a Finite Field
arXiv: Number Theory, 2017Co-Authors: Anju Gupta, Rajendra K. Sharma, Stephen D. CohenAbstract:In this article, we establish a sufficient condition for the existence of a Primitive Element $\alpha \in {\mathbb{F}_{q^n}}$ such that the Element $\alpha+\alpha^{-1}$ is also a Primitive Element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains such Primitive Elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\geq5.$
-
On Primitive normal Elements over finite fields
Asian-european Journal of Mathematics, 2017Co-Authors: Anju, Rajendra K. SharmaAbstract:Let 𝔽qn be an extension of the field 𝔽q of degree n, where q = pk for some positive integer k and prime p. In this paper, we establish a sufficient condition for the existence of a Primitive Element α ∈ 𝔽qn such that α2 + α + 1 is also Primitive as well as a Primitive normal Element α of 𝔽qn over 𝔽q such that α2 + α + 1 is Primitive.
-
Existence of some special Primitive normal Elements over finite fields
Finite Fields and Their Applications, 2017Co-Authors: Rajendra K. SharmaAbstract:In this article, we establish a sufficient condition for the existence of a Primitive Element Fq such that for any matrix (abc0de)M23(Fq) of rank 2, the Element (a2+b+c)/(d+e) is a Primitive Element of Fq, where q=2k for some positive integer k. We also give a sufficient condition for the existence of a Primitive normal Element Fqn over Fq such that (a2+b+c)/(d+e) is a Primitive Element of Fqn for every matrix (abc0de)M23(Fqn) of rank 2.
Dirk Hachenberger - One of the best experts on this subject based on the ideXlab platform.
-
Computational Results on the Existence of Primitive Complete Normal Basis Generators
arXiv: Number Theory, 2019Co-Authors: Dirk Hachenberger, Stefan HackenbergAbstract:We present computational results which strongly support a conjecture of Morgan and Mullen (1996), which states that for every extension $E/F$ of Galois fields there exists a Primitive Element of $E$ which is completely normal over $F$.
-
Primitive complete normal bases: Existence in certain 2-power extensions and lower bounds
Discrete Mathematics, 2010Co-Authors: Dirk HachenbergerAbstract:The present paper is a continuation of the author's work (Hachenberger (2001) [3]) on primitivity and complete normality. For certain 2-power extensions E over a Galois field F"q, we are going to establish the existence of a Primitive Element which simultaneously generates a normal basis over every intermediate field of E/F"q. The main result is as follows: Letq=3mod4and letm(q)>=3be the largest integer such that2^m^(^q^)dividesq^2-1; ifE=F"q"^"2"^"^"^"l, wherel>=m(q)+3, then there exists a Primitive Element inEthat is completely normal overF"q. Our method not only shows existence but also gives a fairly large lower bound on the number of Primitive completely normal Elements. In the above case this number is at least 4@?(q-1)^2^^^l^^^-^^^2. We are further going to discuss lower bounds on the number of such Elements in r-power extensions, where r=2 and q=1mod4, or where r is an odd prime, or where r is equal to the characteristic of the underlying field.
-
Primitivity, freeness, norm and trace
Discrete Mathematics, 2000Co-Authors: Stephen D. Cohen, Dirk HachenbergerAbstract:Given the extension E/F of Galois fields, where F = GF(q) and E = GF(q^n), we prove that, for any Primitive b Element of F*, there exists a Primitive Element in E which is free over F and whose (E, F)-norm is equal to b. Furthermore, if (q,n) unequal (3,2), we prove that, for any nonzero b Element of F, there exists an Element in E which is free over F and whose (E,F)-norm is equal to b. A preliminary investigation of the question of determining whether, in searching for a Primitive Element in E that is free over F, both the (E,F)-norm and the (E,F)-trace can be prescribed is also made: this is so whenever n>=9.
-
Primitivity, freeness, norm and trace
Discrete Mathematics, 2000Co-Authors: Stephen D. Cohen, Dirk HachenbergerAbstract:AbstractGiven the extension E/F of Galois fields, where F=GF(q) and E=GF(qn), we prove that, for any Primitive b∈F∗, there exists a Primitive Element in E which is free over F and whose (E,F)-norm is equal to b. Furthermore, if (q,n)≠(3,2), we prove that, for any nonzero b∈F, there exists an Element in E which is free over F and whose (E,F)-norm is equal to b. A preliminary investigation of the question of determining whether, in searching for a Primitive Element in E that is free over F, both the (E,F)-norm and the (E,F)-trace can be prescribed is also made: this is so whenever n⩾9
Anju Gupta - One of the best experts on this subject based on the ideXlab platform.
-
Primitive Element Pairs with a Prescribed Trace in the Quartic Extension of a Finite Field.
Journal of Algebra and Its Applications, 2020Co-Authors: Stephen D. Cohen, Anju GuptaAbstract:In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a Primitive Element $\alpha $ such that the Element $\alpha+\alpha^{-1}$ is also a Primitive Element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.
-
Pair of Primitive Elements with prescribed traces over finite fields
Communications in Algebra, 2019Co-Authors: Rajendra K. Sharma, Anju GuptaAbstract:In this article, we establish a sufficient condition for the existence of a Primitive Element α∈Fqn such that for any matrix (abc0de)∈M2×3(Fqn) of rank 2, the Element (aα2+bα+c)/(dα+e) is a primiti...
-
Primitive Element pairs with one prescribed trace over a finite field
Finite Fields and Their Applications, 2018Co-Authors: Anju Gupta, Rajendra K. Sharma, Stephen D. CohenAbstract:Abstract In this article, we establish a sufficient condition for the existence of a Primitive Element α ∈ F q n such that the Element α + α − 1 is also a Primitive Element of F q n , and T r F q n | F q ( α ) = a for any prescribed a ∈ F q , where q = p k for some prime p and positive integer k. We prove that every finite field F q n ( n ≥ 5 ) , contains such Primitive Elements except for finitely many values of q and n. Indeed, by computation, we conclude that there are no actual exceptional pairs ( q , n ) for n ≥ 5 .
-
Primitive Element Pairs with One Prescribed Trace over a Finite Field
arXiv: Number Theory, 2017Co-Authors: Anju Gupta, Rajendra K. Sharma, Stephen D. CohenAbstract:In this article, we establish a sufficient condition for the existence of a Primitive Element $\alpha \in {\mathbb{F}_{q^n}}$ such that the Element $\alpha+\alpha^{-1}$ is also a Primitive Element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains such Primitive Elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\geq5.$
Toufik Zaïmi - One of the best experts on this subject based on the ideXlab platform.
-
CM fields with a reciprocal unit-Primitive Element
Comptes Rendus Mathematique, 2018Co-Authors: Cornelius Greither, Toufik ZaïmiAbstract:Abstract Let K be a noncyclotomic CM field. We show that the field K ∩ R has a reciprocal unit-Primitive Element when K does. Also, we prove some related conditions that make the converse of this assertion true.
-
CM FIELDS WITHOUT UNIT-Primitive ElementS
Bulletin of the Australian Mathematical Society, 2017Co-Authors: Cornelius Greither, Toufik ZaïmiAbstract:This is an addendum to a recent paper by Zaïmi, Bertin and Aljouiee [‘On number fields without a unit Primitive Element’, Bull. Aust. Math. Soc.93 (2016), 420–432], giving the answer to a question asked in that paper, together with some historical connections.
-
On number fields without a unit Primitive Element
Bulletin of the Australian Mathematical Society, 2016Co-Authors: Toufik Zaïmi, Marie José Bertin, A. M. AljouieeAbstract:We characterise number fields without a unit Primitive Element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$ , with degree $2d$ where $d$ is odd, and having a unit Primitive Element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$ .
