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Stephen D. Cohen

Publications and source records attributed to Stephen D. Cohen.

At least 19 recordsLinked to original sources
arXiv2026

Consecutive non-square non-primitive pairs in a finite field

Let $q$ be an odd prime power and write \[ θ_q := \frac{ϕ(q-1)}{q-1}. \] If $θ_q < \tfrac{1}{3}$, or if $θ_q = \tfrac{1}{3}$ and $q \notin \{7,13,19,25,37\}$, then the finite field $\F$ contains a pair of consecutive elements that are both non-square and non-primitive. This extends a result of Jarso and Trudgian for prime fields $\Fp$, where the same conclusion was obtained under the stronger condition $θ_p \le \tfrac{1}{4}$. More generally, let $\ell$ be the least odd prime divisor of $q-1$. If $θ_q \le \tfrac{1}{3}$, then $\F$ contains a pair of consecutive elements that are non-squares and $\ell$th powers, with the sole exceptions $q \in \{7,13,19,25,37,43\}$.

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arXiv2025

Triples and quadruples of consecutive squares or non-squares in a finite field

Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ 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 Monzingo(1985) to non-prime fields. Curiously, the same machinery alows the evaluation of the number of consecutive quadruples $\{\al -1, \al,\al+1, \al +2\}$ of square and non-squares over $\F$, when $q$ is a power of 5.

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arXiv2025

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).

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arXiv2023

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 $γ-1,γ,γ+1$ of non-square elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square elements whenever $\FQ$ contains more than 9 elements.

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arXiv2022

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 $ξ\in \mathbb{F}_{q^{n}}$ such that $ξ+ ξ^{-1}$ is primitive and $T(ξ)=a, T(ξ^{-1})=b$, where $T(ξ)$ denotes the trace of $ξ$ 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 $ξ\in \mathbb{F}_{q^{3}}$ such that $ξ+ ξ^{-1}$ is also a primitive element of $\mathbb{F}_{q^{3}}$, and for which the trace of $ξ$ 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.

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arXiv2022

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 $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ 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 $n\ge3$.

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arXiv2022

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 $θ\in \mathbb F_q$ for which $f(θ)$ is $(r,n)$-free and $F(θ)$ 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.

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arXiv2021

The trace of primitive and $2$-primitive elements in finite fields, revisited

By definition primitive and $2$-primitive elements of a finite field extension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primitive element $ξ\in \mathbb{F}_{q^n}$ with prescribed trace in the ground field $\mathbb{F}_q$. Here we amend our previous proofs of these results, firstly, by a reduction of these problems to extensions of prime degree $n$ and, secondly, by deriving an exact expression for the number of squares in $\mathbb{F}_{q^n}$ whose trace has prescribed value in $\mathbb{F}_q$. The latter corrects an error in the proof in the case of $2$-primitive elements. We also streamline the necessary computations.

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arXiv2021

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 $n$.

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arXiv2019

The translate and line properties for 2-primitive elements in quadratic extensions

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements if, for every $α,θ\in\mathbb{F}_{q^n}^*$, such that $\mathbb{F}_{q^n}=\mathbb{F}_q(θ)$, there exists some $x\in\mathbb{F}_q$, such that $α(θ+x)$ has multiplicative order $(q^n-1)/r$. Likewise, if, in the above definition, $α$ is restricted to the value $1$, we say that $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the translate property. In this paper we take $r=n=2$ (so that necessarily $q$ is odd) and prove that $\mathbb{F}_{q^2} /\mathbb{F}_q$ possesses the translate property for 2-primitive elements unless $q \in \{5,7,11,13,31,41\}$. With some additional theoretical and computational effort, we show also that $\mathbb{F}_{q^2} /\mathbb{F}_q$ possesses the line property for 2-primitive elements unless $q \in \{3,5,7,9,11,13,31,41\}$.

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arXiv2019

Finite field extensions with the line or translate property for $r$-primitive elements

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements property if, for every $α,θ\in\mathbb{F}_{q^n}^*$, such that $\mathbb{F}_{q^n}=\mathbb{F}_q(θ)$, there exists some $x\in\mathbb{F}_q$, such that $α(θ+x)$ has multiplicative order $(q^n-1)/r$. We prove that, for sufficiently large prime powers $q$, $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements. We also discuss the (weaker) translate property for extensions.

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arXiv2019

Primitive values of rational functions at primitive elements of a finite field

Given a prime power $q$ and an integer $n\geq2$, we establish a sufficient condition for the existence of a primitive pair $(α,f(α))$ where $α\in \mathbb{F}_q$ and $f(x) \in \mathbb{F}_q(x)$ is a rational function of degree $n$. (Here $f=f_1/f_2$, where $f_1, f_2$ are coprime polynomials of degree $n_1,n_2$, respectively, and $n_1+n_2=n$.) For any $n$, such a pair is guaranteed to exist for sufficiently large $q$. Indeed, when $n=2$, such a pair definitely does {\em not} exist only for 28 values of $q$ and possibly (but unlikely) only for at most $3911$ other values of $q$.

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arXiv2019

Primitive Element Pairs with a Prescribed Trace in the Quartic Extension of a Finite Field

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 $α$ such that the element $α+α^{-1}$ is also a primitive element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(α)=a$ 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.

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arXiv2018

Existence results for primitive elements in cubic and quartic extensions of a finite field

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $γ$ generates $\Fqn$ over $\Fq$ and $β$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $β(γ+ a)$ 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 $n=4$.

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arXiv2018

Primitive Element Pairs with One Prescribed Trace over a Finite Field

In this article, we establish a sufficient condition for the existence of a primitive element $α\in {\mathbb{F}_{q^n}}$ such that the element $α+α^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(α)=a$ 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 $n\geq5.$

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arXiv2018

Primitive values of quadratic polynomials in a finite field

We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field $\mathbb{F}_{q}$ such that $Q(g)$ is also a primitive root, where $Q(x)= ax^2 + bx + c$ is a quadratic polynomial with $a, b, c\in \mathbb{F}_{q}$ such that $b^{2} - 4ac \neq 0$.

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arXiv2017

Lehmer numbers and primitive roots modulo a prime

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the known expression for the number of Lehmer numbers modulo $p$ and improve the Wang--Wang estimate for the number of solutions to the Golomb--Lehmer primitive root problem.

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