Related papers: Finite field extensions with the line or translate…

Let $q$ be a prime power and $n$ and $r$ be positive integers. It is well known that the linearized binomial $L_r(x)=x^{q^r}+ax\in\mathbb{F}_{q^n}[x]$ is a permutation polynomial if and only if $(-1)^{n/d}a^{{(q^n-1)}/{(q^{d}-1)}}\neq 1$…

V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

Let F_q be the finite field with cardinality q, where q is a prime power. Given a finite field extension F_q^n over F_q and a,b in (F_q)^{*}, we investigate in this article the number N_n(a,b) of elements in F_q^n whose norm equals a and…

Let F_q be the finite field with q elements. Consider the standard embedding GL(n,F_q) -> GL(n+1,F_q). In this paper we prove that for every irreducible representation pi of GL(n+1,F_q) over algebraically closed fields of characteristic…

After extending the classic notion of a tight Heffter array H$(m,n)$ to any group of order $2mn+1$, we give direct constructions for elementary abelian tight Heffter arrays, hence in particular for prime tight Heffter arrays. If $q=2mn+1$…

The main result of the present article is a proof of Fermat's Last Theorem for sufficiently large prime exponents $p$ with $p \equiv 2 \pmod{3}$ over certain number fields. A particular case of these fields are the maximal real subfields of…

For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…

We construct infinite Galois extensions $K$ of $\mathbb{Q}$ that satisfy the Northcott property on elements of small height, and where this property can be deduced solely from the splitting behavior of prime numbers in $K$. We also give…

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n = (q+1)/2$. For each $s \in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y): = y^n + (1-y)^n -s$, and we give a necessary and…

Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over…

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the…

Let K denote a finite extension of Qp. We give necessary and sufficient conditions for an infinite totally wildly ramified extension L/K to be strictly APF in the sense of Fontaine-Wintenberger. Our conditions are phrased in terms of the…

Let $(W, R)$ be a Coxeter system and let $w \in W$. We say that $u$ is a prefix of $w$ if there is a reduced expression for $u$ that can be extended to one for $w$. That is, $w = uv$ for some $v$ in $W$ such that $\ell(w) = \ell(u) +…

Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements, which satisfies $|E| > q$. There exist $x,y \in E$ such that $|E \cdot (y-x)| > q/2.$ In particular, $(E+E) \cdot (E-E) =…

In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.

Let $f=a_0+ a_{1}x+\cdots+a_m x^m\in \Bbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $\alpha$ such that $|a_m| \alpha^m>|a_0|+|a_1|\alpha+\cdots+|a_{m-1}|\alpha^{m-1}$. We prove that if there exist…

For integer $q$, let $\chi$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}\Lambda(n)\chi(n+a),$$ where $\Lambda(n)$ is the von Mangoldt function. This…

In the paper we study finitely generated linear groups of finite rank which have faithful irreducible primitive representations over a field of characteristic zero. We prove that if an infinite finitely generated linear group $G$ of finite…

We adapt the proof of the Green-Tao theorem on arithmetic progressions in primes to the setting of polynomials over a finite field, to show that for every $k$, the irreducible polynomials in $\mathbf{F}_q[t]$ contain configurations of the…