Related papers: Further results on the Morgan-Mullen conjecture

Let $n\ge 2$ be an integer and let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given $\mathbb F_q$-affine hyperplanes $\mathcal A_1, \ldots, \mathcal A_n$ of $\mathbb F_{q^n}$ in general position, we…

Given positive integers $q,n,m$ and $a\in\mathbb{F}_{q}$, where $q$ is an odd prime power and $n\geq 5$, we investigate the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ such…

In this paper we derive a formula for the number of $N$-free elements over a finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula…

Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of primitive normal pairs of the…

We introduce {\em admissible collections} for a finite group $G$ and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the {\em Quillen dimension at $p$ property}, a strong version of…

We say a natural number~$n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces…

For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^{n}}$ over…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive divisor $r$ of $q^n-1$, the element $\alpha \in \mathbb{F}_{q^n}^*$ is called \textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a…

We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers.

We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular,…

We extend the Main Theorem of Aschbacher and Smith on Quillen's Conjecture from $p>5$ to the remaining odd primes $p = 3,5$. In the process, we develop further combinatorial and homotopical methods for studying the poset of nontrivial…

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…

Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $\epsilon \in…

Let $n$ be a positive integer and let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2,…

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from $\mathbb{F}_q^n$ to $\mathbb{F}_q$ to the set of affine functions from $\mathbb{F}_q^n$ to $\mathbb{F}_q$. We prove…

By using Andrews's explicit formulae of the $q$-Fibonacci sequence introduced by Schur, we prove certain congruences of the $q$-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that…