Related papers: Frobenius matrices and a variant of Zolotarev's th…
In this paper, we study certain determinants over finite fields. Let $\mathbb{F}_q$ be the finite field of $q$ elements and let $a_1,a_2,\cdots,a_{q-1}$ be all nonzero elements of $\mathbb{F}_q$. Let…
It is well known that there exists a significant equivalence between the vector space $\mathbb{F}_{q}^n$ and the finite fields $\mathbb{F}_{q^n}$, and many scholars often view them as the same in most contexts. However, the precise…
Frobenius' Theorem states that the algebra of quaternions $\mathbb H$ is, besides the fields of real and complex numbers, the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then…
In this paper, we establish an analogue of the Fundamental Theorem of Algebra for polynomial matrix equations, where both the coefficient matrices and the unknown matrix are $Q$-circulant matrices. This result generalizes Abramov's result…
Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_q$ and finite cyclic rings $\mathbb{Z}/p^r\mathbb{Z}$, in the…
By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure ${{\bar \BF}}_q$ of the finite field $\BF_q$ of $q$ elements, we establish a relation between the representation theory of $A$ over ${{\bar…
Let $R$ be a standard graded finitely generated algebra over an $F$-finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that that Frobenius complexity of $R$ is finite. Moreover, we…
Frobenius built a representation theory of finite groups in the process of obtaining the irreducible factorization of the group determinant. Here, we give a generalization of Frobenius' theorem. The generalization leads to a corollary on…
Let $\mathbb{F}_q$ be the finite field of $q$ elements and $a_1,a_2, \ldots, a_k, b\in \mathbb{F}_q$. We investigate $N_{\mathbb{F}_q}(a_1, a_2, \ldots,a_k;b)$, the number of ordered solutions $(x_1, x_2, \ldots,x_k)\in\mathbb{F}_q^k$ of…
We generalize the concept of a number derivative, and examine one particular instance of a deformed number derivative for finite field elements. We find that the derivative is linear when the deformation is a Frobenius map and go on to…
In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting $a_1,\cdots,a_{(q-1)/2}$ be all non-zero…
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…
We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.
In this short note, we introduce an analogue of Wilson's theorem for all nonzero elements $a_1,a_2,...,a_{q-1}$ of a finite filed $\mathbb{F}$ with $|\mathbb{F}|=q\geq 3$, as follows: $$ \sum_{1\leq i_1< i_2<...< i_k\leq…
Chebotarev's theorem on roots of unity states that all minors of the Fourier matrix of prime size are non-vanishing. This result has been rediscovered several times and proved via different techniques. We follow the proof of Evans and…
In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.
We prove some cases of a conjecture of Lewis, Reiner and Stanton regarding Hilbert series corresponding to the action of $Gl_n(\mathbb{F}_q)$ on a polynomial ring modulo Frobenius powers. We also give a few conjectures about the invariant…
Let $\mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with…
By the Fourier transformations, any group-invariant functions over finite Abelian groups are transformed into group-invariant functions over the character groups. In this paper, we calculate matrix elements of this transformations under…
We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes…