Abstract:
A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each conjugacy class is big enough. The result is applied to the problem on the number of irreducible components of the Hurwitz space of degree $d$ marked coverings of $\mathbb P^1$ with given Galois group $G$ and fixed collection of local monodromies.

Abstract:
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise description of quotients of skew polynomial rings by a left principal ideal, using results relating skew polynomial rings to Azumaya algebras. We use this description to give a new factorization algorithm for skew polynomials, and to give other algorithms related to factorizations of skew polynomials, like counting the number of factorizations as a product of irreducibles.

Abstract:
We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter subgroups, we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

Abstract:
The Brauer monoid is studied by the notion of 0-cohomology. We investigate the impact of invertible elements of modifications on the structure of the Brauer monoid, especially for finite fields.

Abstract:
In this note, we first discuss some properties of generated $\sigma$-fields and a simple approach to the construction of finite $\sigma$-fields. It is shown that the $\sigma$-field generated by a finite class of $\sigma$-distinct sets which are also atoms, is the same as the one generated by the partition induced by them. The range of the cardinality of such a generated $\sigma$-field is explicitly obtained. Some typical examples and their complete forms are discussed. We discuss also a simple algorithm to find the exact cardinality of some particular finite $\sigma$-fields. Finally, an application of our results to statistics, with regard to independence of events, is pointed out.

Abstract:
Looking forward to introducing an analysis in Galois Fields, discrete functions are considered (such as transcendental ones) and MacLaurin series are derived by Lagrange's Interpolation. A new derivative over finite fields is defined which is based on the Hasse Derivative and is referred to as negacyclic Hasse derivative. Finite field Taylor series and alpha-adic expansions over GF(p), p prime, are then considered. Applications to exponential and trigonometric functions are presented. Theses tools can be useful in areas such as coding theory and digital signal processing.

Abstract:
Finite fields form an important chapter in abstract algebra, and mathematics in general, yet the traditional expositions, part of Abstract Algebra courses, focus on the axiomatic presentation, while Ramification Theory in Algebraic Number Theory, making a suited topic for their applications, is usually a separated course. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience, and bridging the above mentioned gap. Such lattice models of finite fields provide a good basis for later developing their study in a more concrete way, including decomposition of primes in number fields, Frobenius elements, and Frobenius lifts, allowing to approach more advanced topics, such as Artin reciprocity law and Weil Conjectures, while keeping the exposition to the concrete level of familiar number systems. Examples are provided, intended for an undergraduate audience in the first place.

Abstract:
We establish a connection between finite fields and finite dynamical systems. We show how this connection can be used to shed light on some problems in finite dynamical systems and in particular, in linear systems.

Abstract:
This is an introduction to the theory of normal bases of finite fields. The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given. The last few chapters cover the theory of guassian and period normal bases of finite fields of low degrees. The last chapter presents the asymptotic proofs of the existence of primitive polynomials of degree n with approximately n/2 arbitrary coefficients, and primitive normal polynomials of arbitrary traces.

Abstract:
We generalize Burgess' results on partial Gaussian sums to arbitrary finite fields. The main ingredients are the classical method of amplification, two deep results on multiplicative energy for subsets in finite fields which are obtained respectively by the tools from additive combinatorics and geometry of numbers, and a technique of Chamizo for treating the difficulty caused by additive character. Our results include the recent works on character sums in finite fields by M.-C. Chang and S. V. Konyagin.