SCStephen D. Cohen
Stephen D. Cohen
*This is an algorithmically generated author record based on publicly available information.
Research Subjects
Research Interests
Stephen D. Cohen has a total of 126 co-authors. Here are the top co-authors by number of shared publications:
- Giorgos Kapetanakis
University of Thessaly
7 shared publications - Tim Trudgian
Univ New South Wales, Sch Sci, Canberra, ACT, Australia
7 shared publications - Tomas Oliveira Silva
University of Aveiro
6 shared publications - Andrew R. Booker
University of Bristol
4 shared publications Gary L. Mullen
4 shared publications- Sophie Huczynska
School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, UK
4 shared publications
Latest Publications
- 8 October 2026
- Published by World Scientific Pub Co Pte Ltd in Journal of Algebra and Its Applications
Triples and quadruples of consecutive squares or non-squares in a finite field
Let [Formula: see text] be the finite field of odd prime power order [Formula: see text]. We find explicit expressions for the number of triples [Formula: see text] of consecutive nonzero squares in [Formula: see text] and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of [M. G. Monzingo, On the distribution of consecutive triples of quadratic residues and quadratic nonresidues and related topics, Fibonacci Quart. 23 (1985) 133–138] to non-prime fields. Curiously, the same machinery allows the evaluation of the number of consecutive quadruples [Formula: see text] of square and non-squares over [Formula: see text], when [Formula: see text] is a power of 5.
- 8 October 2026
- Published by World Scientific Pub Co Pte Ltd in Journal of Algebra and Its Applications
Triples and quadruples of consecutive squares or non-squares in a finite field
Let [Formula: see text] be the finite field of odd prime power order [Formula: see text]. We find explicit expressions for the number of triples [Formula: see text] of consecutive nonzero squares in [Formula: see text] and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of [M. G. Monzingo, On the distribution of consecutive triples of quadratic residues and quadratic nonresidues and related topics, Fibonacci Quart. 23 (1985) 133–138] to non-prime fields. Curiously, the same machinery allows the evaluation of the number of consecutive quadruples [Formula: see text] of square and non-squares over [Formula: see text], when [Formula: see text] is a power of 5.
- Open Access
Finite fields whose members are the sum of a potent and a 4-potent
We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs $(q,n)$ for which this is the case. This continues a recent publication by Cohen-Danchev et al. in Turk. J. Math. (2024) in which the tripotent version was examined in-depth as well as it extends recent results of this branch established by Abyzov-Tapkin in Sib. Math. J. (2024).Open AccessFinite fields whose members are the sum of a potent and a 4-potent
We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs $(q,n)$ for which this is the case. This continues a recent publication by Cohen-Danchev et al. in Turk. J. Math. (2024) in which the tripotent version was examined in-depth as well as it extends recent results of this branch established by Abyzov-Tapkin in Sib. Math. J. (2024). - Open Access
Rings and finite fields whose elements are sums or differences of tripotents and potents
We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either exactly seven or strictly more than nine elements, each matrix over $F$ is presentable as a sum of of a tripotent matrix and a $q$-potent matrix if and only if each element in $F$ is presentable as a sum of a tripotent and a $q$-potent, whenever $q>1$ is an odd integer. In addition, if $Q$ is a power of an odd prime and $F$ is a field of odd characteristic, having cardinality strictly greater than $9$, then, for all $n\geq 1$, the matrix ring $\mathbb{M}_n(F)$ is weakly $(Q-1)$-torsion clean if and only if $F$ is a finite field of cardinality $Q$. A novel contribution to the ring-theoretical theme of this study is the classification of finite fields $\FQ$ of odd order in which every element is the sum of a tripotent and a potent. In this regard, we obtain an expression for the number of consecutive triples $\gamma-1,\gamma,\gamma+1$ of non-square elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square elements whenever $\FQ$ contains more than 9 elements.Open AccessRings and finite fields whose elements are sums or differences of tripotents and potents
We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either exactly seven or strictly more than nine elements, each matrix over $F$ is presentable as a sum of of a tripotent matrix and a $q$-potent matrix if and only if each element in $F$ is presentable as a sum of a tripotent and a $q$-potent, whenever $q>1$ is an odd integer. In addition, if $Q$ is a power of an odd prime and $F$ is a field of odd characteristic, having cardinality strictly greater than $9$, then, for all $n\geq 1$, the matrix ring $\mathbb{M}_n(F)$ is weakly $(Q-1)$-torsion clean if and only if $F$ is a finite field of cardinality $Q$. A novel contribution to the ring-theoretical theme of this study is the classification of finite fields $\FQ$ of odd order in which every element is the sum of a tripotent and a potent. In this regard, we obtain an expression for the number of consecutive triples $\gamma-1,\gamma,\gamma+1$ of non-square elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square elements whenever $\FQ$ contains more than 9 elements. - 22 June 2022
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathematique
- Vol. 360 (G6) , 641-652
The existence of -primitive points on curves using freenessLet be the cyclic group of order , a divisor of and a divisor of . We introduce the set of -free elements of and derive a lower bound for the number of elements for which is -free and is -free, where . As an application, we consider the existence of -primitive points on curves like and find, in particular, all the odd prime powers for which the elliptic curves contain an -primitive point.- 22 June 2022
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathematique
- Vol. 360 (G6) , 641-652
The existence of -primitive points on curves using freenessLet be the cyclic group of order , a divisor of and a divisor of . We introduce the set of -free elements of and derive a lower bound for the number of elements for which is -free and is -free, where . As an application, we consider the existence of -primitive points on curves like and find, in particular, all the odd prime powers for which the elliptic curves contain an -primitive point.- 25 April 2022
- Published by Cambridge University Press (CUP) in Bulletin of the Australian Mathematical Society
- Vol. 106 (3) , 458-462
PRIMITIVE ELEMENT PAIRS WITH A PRESCRIBED TRACE IN THE CUBIC EXTENSION OF A FINITE FIELD
We prove that for any prime power $q\notin \{3,4,5\}$ , the cubic extension $\mathbb {F}_{q^{3}}$ of the finite field $\mathbb {F}_{q}$ contains a primitive element $\xi $ such that $\xi +\xi ^{-1}$ is also primitive, and $\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$ for any prescribed $a\in \mathbb {F}_{q}$ . This completes the proof of a conjecture of Guptaet al.[‘Primitive element pairs with one prescribed trace over a finite field’,Finite Fields Appl.54(2018), 1–14] concerning the analogous problem over an extension of arbitrary degree $n\ge 3$ .- 25 April 2022
- Published by Cambridge University Press (CUP) in Bulletin of the Australian Mathematical Society
- Vol. 106 (3) , 458-462
PRIMITIVE ELEMENT PAIRS WITH A PRESCRIBED TRACE IN THE CUBIC EXTENSION OF A FINITE FIELD
We prove that for any prime power $q\notin \{3,4,5\}$ , the cubic extension $\mathbb {F}_{q^{3}}$ of the finite field $\mathbb {F}_{q}$ contains a primitive element $\xi $ such that $\xi +\xi ^{-1}$ is also primitive, and $\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$ for any prescribed $a\in \mathbb {F}_{q}$ . This completes the proof of a conjecture of Guptaet al.[‘Primitive element pairs with one prescribed trace over a finite field’,Finite Fields Appl.54(2018), 1–14] concerning the analogous problem over an extension of arbitrary degree $n\ge 3$ .- Open Access
Primitive elements with prescribed traces
Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero element $\xi \in \mathbb{F}_{q^{n}}$ such that $\xi+ \xi^{-1}$ is primitive and $T(\xi)=a, T(\xi^{-1})=b$, where $T(\xi)$ denotes the trace of $\xi$ in $\mathbb{F}_{q}$. This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of $n \geq 5$. A related study involves the cubic extension $\mathbb{F}_{q^{3}}$ of $\mathbb{F}_{q}$. We show that if $q\geq 8\cdot 10^{12}$ then, for any $a\in \mathbb{F}_{q}$ we can find a primitive element $\xi \in \mathbb{F}_{q^{3}}$ such that $\xi + \xi^{-1}$ is also a primitive element of $\mathbb{F}_{q^{3}}$, and for which the trace of $\xi$ is equal to $a$. The improves a result of Cohen and Gupta. Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions.Open AccessPrimitive elements with prescribed traces
Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero element $\xi \in \mathbb{F}_{q^{n}}$ such that $\xi+ \xi^{-1}$ is primitive and $T(\xi)=a, T(\xi^{-1})=b$, where $T(\xi)$ denotes the trace of $\xi$ in $\mathbb{F}_{q}$. This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of $n \geq 5$. A related study involves the cubic extension $\mathbb{F}_{q^{3}}$ of $\mathbb{F}_{q}$. We show that if $q\geq 8\cdot 10^{12}$ then, for any $a\in \mathbb{F}_{q}$ we can find a primitive element $\xi \in \mathbb{F}_{q^{3}}$ such that $\xi + \xi^{-1}$ is also a primitive element of $\mathbb{F}_{q^{3}}$, and for which the trace of $\xi$ is equal to $a$. The improves a result of Cohen and Gupta. Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions. - Open Access
Primitive element pairs with a prescribed trace in the cubic extension of a finite field
We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.Open AccessPrimitive element pairs with a prescribed trace in the cubic extension of a finite field
We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$. - Open Access
The existence of $\mathbb{F}_q$-primitive points on curves using freeness
Let $\mathcal C_Q$ be the cyclic group of order $Q$, $n$ a divisor of $Q$ and $r$ a divisor of $Q/n$. We introduce the set of $(r,n)$-free elements of $\mathcal C_Q$ and derive a lower bound for the the number of elements $\theta \in \mathbb F_q$ for which $f(\theta)$ is $(r,n)$-free and $F(\theta)$ is $(R,N)$-free, where $ f, F \in \mathbb F_q[x]$. As an application, we consider the existence of $\mathbb F_q$-primitive points on curves like $y^n=f(x)$ and find, in particular, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 \pm x$ contain an $\mathbb F_q$-primitive point.Open AccessThe existence of $\mathbb{F}_q$-primitive points on curves using freeness
Let $\mathcal C_Q$ be the cyclic group of order $Q$, $n$ a divisor of $Q$ and $r$ a divisor of $Q/n$. We introduce the set of $(r,n)$-free elements of $\mathcal C_Q$ and derive a lower bound for the the number of elements $\theta \in \mathbb F_q$ for which $f(\theta)$ is $(r,n)$-free and $F(\theta)$ is $(R,N)$-free, where $ f, F \in \mathbb F_q[x]$. As an application, we consider the existence of $\mathbb F_q$-primitive points on curves like $y^n=f(x)$ and find, in particular, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 \pm x$ contain an $\mathbb F_q$-primitive point. - Open Access
The trace of 2-primitive elements of finite fields (amended version)
Let $q$ be a prime power and $n, r$ integers such that $r\mid q^n-1$. An element of $\mathbb{F}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called \emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a $2$-primitive element of $\mathbb{F}_{q^n}$ with arbitrarily prescribed $\mathbb{F}_q$ trace when $n\geq 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$. A feature of this amended version is the reduction of the discussion to extensions of prime degree $n$.Open AccessThe trace of 2-primitive elements of finite fields (amended version)
Let $q$ be a prime power and $n, r$ integers such that $r\mid q^n-1$. An element of $\mathbb{F}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called \emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a $2$-primitive element of $\mathbb{F}_{q^n}$ with arbitrarily prescribed $\mathbb{F}_q$ trace when $n\geq 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$. A feature of this amended version is the reduction of the discussion to extensions of prime degree $n$. - 15 July 2020
- Published by World Scientific Pub Co Pte Ltd in Journal of Algebra and Its Applications
- Vol. 20 (09)
Primitive element pairs with a prescribed trace in the quartic extension of a finite field
In this paper, we give a largely self-contained proof that the quartic extension [ ] of the finite field [ ] contains a primitive element [ ] such that the element [ ] is also a primitive element of [ ] and [ ] for any prescribed [ ]. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl. 54 (2018) 1–14.].
- 15 July 2020
- Published by World Scientific Pub Co Pte Ltd in Journal of Algebra and Its Applications
- Vol. 20 (09)
Primitive element pairs with a prescribed trace in the quartic extension of a finite field
In this paper, we give a largely self-contained proof that the quartic extension [ ] of the finite field [ ] contains a primitive element [ ] such that the element [ ] is also a primitive element of [ ] and [ ] for any prescribed [ ]. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl. 54 (2018) 1–14.].