On Gauss sums over dedekind domains

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Non-negative integral matrices with given spectral radius and controlled dimension

A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

A note on Dedekind Criterion

Let [Formula: see text] be an algebraic number field with [Formula: see text] an algebraic integer having minimal polynomial [Formula: see text] over the field [Formula: see text] of rational numbers and [Formula: see text] be the ring of algebraic integers of [Formula: see text]. For a fixed prime number [Formula: see text], let [Formula: see text] be the factorization of [Formula: see text] modulo [Formula: see text] as a product of powers of distinct irreducible polynomials over [Formula: see text] with [Formula: see text] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number [Formula: see text] does not divide the index [Formula: see text] if and only if [Formula: see text] is coprime with [Formula: see text] where [Formula: see text]. This criterion has been widely used and generalized. In this paper, a simple proof of Generalized Dedekind Criterion [S. K. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation rings, Comm. Algebra 38 (2010) 684–696] using elementary valuation theory is given.

ON THE TRAINING METHODOLOGY FOR COMPILING SOME TYPE OF ARITHMETIC PROBLEMS

The article is devoted to a current topic related to the development of methods for composing problems in teaching students of pedagogical universities of mathematical faculties. This problem becomes especially important in the context of the need to involve students in independent creative activities to acquire and apply knowledge. The material is presented in relation to the section of the discipline «Algebra and number theory», dedicated to solving Diophantine equations, the main objectives of which are not only mastering the theory and algorithms for solving basic problems, but also obtaining the necessary knowledge and skills for further professional activity. Solving a problem, the student must not only solve it correctly and quickly enough, but also show the creative component of the activity, using it as much as possible for their mathematical development. In this regard, the process of composing problems by students is undoubtedly useful, which reflects the systematic application of the material and elements of mathematical actions based on the laws and methods of mathematics. In addition, the ability to compose problems will be required in future activities related to teaching mathematics. The processes of solving and composing tasks are interconnected and this allows you to increase the efficiency and effectiveness of composing and solving tasks. Therefore, the teacher can give a task to the student with the requirement to compose (fully or partially) and solve the problem. In this paper, examples of tasks for the compilation of indefinite equations solvable in integers are considered, for the solution of which the methods of number theory are used: the study of possible residuals from dividing an algebraic integer expression by a specific integer; finding integer solutions to a linear equation with two variables. The stages of composing Diophantine equations are described in detail, the ways of obtaining equations solvable on a given set of integers or natural numbers are analyzed, and the application of various theoretical propositions used for their solution is shown.