Related papers: Generators of finite fields with prescribed traces

To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and…

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds…

In this paper, we consider the direct and inverse problems of the description of lattice positive random fields by various systems of finite-dimensional (as well as one-point) probability distributions parameterized by boundary conditions.…

We give a characterization of ramification groups of local fields with imperfect residue fields, using those for local fields with perfect residue fields. As an application, we reprove an equality of ramification groups for abelian…

Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the…

We introduce totally nonnegative Grassmannians over finite fields where an element of a finite field is nonnegative if it is a square of an element of the finite field. Explicit point counts are given in some special cases where we find new…

It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the…

Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show that there are infinitely many number fields of degree $D$ whose primitive elements all have relatively large height in terms of $b$, $D$ and the discriminant of…

We prove that if $G$ is a finite primitive permutation group and if $g$ is an element of $G$, then either $g$ has a cycle of length equal to its order, or for some $r$, $m$ and $k$, the group $G \leq \mathrm{Sym}(m) \textrm{wr}…

Using the properties of the ideal of the coordinate Hermite interpolation on n-dimensional grid [4], we prove that the extension k in k[x1, x2, ..., xn] / (f1(x1), ..., fn(xn)) has a primitive element if and only if at most one of the…

This is an expository note discussing how the Erdos--Ramanujan proof of Bertrand's postulate may be adapted to show the existence of finite fields.

In recent decades, the defect of finite extensions of valued fields has emerged as the main obstacle in several fundamental problems in algebraic geometry such as the local uniformization problem. Hence, it is important to identify…

In this note, we presented a new decomposition of elements of finite fields of even order and illustrated that it is an effective tool in evaluation of some specific exponential sums over finite fields, the explicit value of some…

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads…

In this work, we investigate additive complementary dual (ACD) codes and their construction over finite fields $\mathbb{F}_{q^2}$ with respect to the trace inner products, where $q$ is a prime power. First, we associate an additive code…

In this article, we study several problems related to virtual traces for finite group actions on schemes of finite type over an algebraically closed field. We also discuss applications to fixed point sets. Our results generalize previous…

The structure of groups for which certain sets of commutator subgroups are finite is investigated, with a particular focus on the relationship between these groups and those with finite derived subgroup.

Magnetic fields have been observed in galaxies, clusters of galaxies and probably in superclusters. While mechanisms exist to generate these in the late universe, it is possible that magnetic fields have existed since very early times. This…

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…

An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field; the polynomial generators of constacyclic codes of length $\ell^tp^s$ are characterized, where $p$ is the characteristic of the finite…