Trivial units for group rings over rings of algebraic integers

Let be a nontrivial torsion group and be the ring of integers of an algebraic number field. The necessary and sufficient conditions are given under which has only trivial units.

     

Hochschild homology of rings of algebraic integers

The purpose of this paper is to remind that the André-Quillen homology theory is a very useful tool to study Hochschild homology of commutative algebras. We choose as an example a computation of Hochschild homology of Dedekind extensions.     

Conjugate algebraic integers in real point sets

Mathematische Zeitschrift -     

Gauss sums and Kloosterman sums over residue rings of algebraic integers

Let denote the ring of integers of an algebraic number field of degree which is totally and tamely ramified at the prime . Write , where . We evaluate the twisted Kloosterman sum

where and denote trace and norm, and where is a Dirichlet character (mod ). This extends results of Salié for and of Yangbo Ye for prime dividing Our method is based upon our evaluation of the Gauss sum

which extends results of Mauclaire for .

     

Normsets and determination of unique factorization in rings of algebraic integers

The image of the norm map from to (two rings of algebraic integers) is a multiplicative monoid . We present conditions under which is a UFD if and only if has unique factorization into irreducible elements. From this we derive a bound for checking if is a UFD.

     

Diameters of complete sets of conjugate algebraic integers of small degree

We give bounds for the coefficients of a polynomial as functions of the diameter of its roots, hence we obtain polynomials with minimal diameters and small degree

     

Bases for sets of integers

We are interested in expressing each of a given set of non-negative integers as the sum of two members of a second set, the second set to be chosen as economically as possible.So let us call B a basis for A if to every a ∈ A there exist b, b′ ∈ B such that a = b + b′. We concern ourselves primarily with finite sets, A, since the results for infinite sets generally follow from these by the familiar process of condensation.     

Size of algebraic integers

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 152–154, April, 1991.     

On small sets of integers

The Ramanujan Journal - An upper quasi-density on $$\mathbf{{H}}$$ (the integers or the non-negative integers) is a real-valued subadditive function $$\mu ^\star $$ defined on the whole power set...     

On smooth sets of integers

This work studies evenly distributed sets of integers—sets whose quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation. Namely, for ρ,Δ∈R a set A of integers is (ρ,Δ)- smooth if for any interval I of integers; a set A is Δ-smooth if it is (ρ,Δ)-smooth for some real number ρ. The paper introduces the concept of Δ-smooth sets and studies their mathematical structure. It focuses on tools for constructing smooth sets having certain desirable properties and, in particular, on mathematical operations on these sets. Three additional papers by us are build on the work of this paper and present practical applications of smooth sets to common and well-studied scheduling problems.One of the above mathematical operations is composition of sets of natural numbers. For two infinite sets A,B⊆N, the composition of A and B is the subset D of A such that, for all i, the ith member of A is in D if and only if the ith member of N is in B. This operator enables the partition of a (ρ,Δ)-smooth set into two sets that are (ρ₁,Δ)-smooth and (ρ₂,Δ)-smooth, for any ρ₁,ρ₂ and Δ obeying some reasonable restrictions. Another powerful tool for constructing smooth sets is a one-to-one partial function f from the unit interval into the natural numbers having the property that any real interval X⊆[0,1) has a subinterval Y which is ‘very close’ to X s.t. f(Y) is (ρ,Δ)-smooth, where ρ is the length of Y and Δ is a small constant.