Related papers: Bounds for the relative class number problem for f…
We give a function field specific, algebraic proof of the main results of class field theory for abelian extensions of degree coprime to the characteristic. By adapting some methods known for number fields and combining them in a new way,…
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
One of the traditional applications of relation algebras is to provide a setting for infinite-domain constraint satisfaction problems. Complexity classification for these computational problems has been one of the major open research…
Using class field theory, we prove a restriction on the intersection of the maximal abelian extensions associated with different number fields. This restriction is then used to improve a result of Rosen and Silverman about the linear…
We calculate the number of the isomorphism class of the finite flat models over the ring of integers of an absolutely ramified $p$-adic field of constant group schemes of rank two over finite fields, by counting the rational points of a…
We introduce the notion of residual finiteness for categories. In analogy with the group-theoretic setting, we prove that free categories and finitely generated subcategories of finite-dimensional vector spaces are residually finite.…
In this paper, we introduce the commutativity degree of a finite-dimensional Lie algebra over a finite field and determine upper and lower bounds for it. Moreover, we study some relations between the notion of commutativity degree and known…
The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.
We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…
It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields…
It is known that infinitely many number fields and function fields of any degree $m$ have class number divisible by a given integer $n$. However, significantly less is known about the indivisibility of class numbers of such fields. While…
This article surveys recent advances and future challenges in the $2$-representation theory of finitary $2$-categories with a particular emphasis on problems related to classification of various classes of $2$-representations.
In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds…
We study a class of fractional $p$-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related…
Pippenger's Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set $A$ and taking values in a possibly different set $B$, where any…
We consider the problem of determining, given x, y in Z^k and a finite set F of affine functions on Z^k, whether y is reachable from x by applying the functions F. We also consider the analogous problem over N^k. These problems are known to…
We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of…
The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. The special attention is given to modelling such functions by systems of functional…
In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.
Abstrct: In this note, by considering fractionally linear functions over a finite field and consequently developing an abstract sequence, we study some of its properties.