Set Of Algebraic Integers Research Articles

Primes in denominators of algebraic numbers

Denote the set of algebraic numbers as [Formula: see text] and the set of algebraic integers as [Formula: see text]. For [Formula: see text], consider its irreducible polynomial in [Formula: see text], [Formula: see text]. Denote [Formula: see text]. Drungilas, Dubickas and Jankauskas show in a recent paper that [Formula: see text]. Given a number field [Formula: see text] and [Formula: see text], we show that there is a subset [Formula: see text], for which [Formula: see text]. We prove that [Formula: see text] is a principal ideal domain if and only if the primes in [Formula: see text] generate the class group of [Formula: see text]. We show that given [Formula: see text], we can find a finite set [Formula: see text], such that for every number field [Formula: see text], we have [Formula: see text]. We study how this set [Formula: see text] relates to the ring [Formula: see text] and the ideal [Formula: see text] of [Formula: see text]. We also show that [Formula: see text] satisfy [Formula: see text] if and only if [Formula: see text] for all number fields [Formula: see text].

Structural results for semigroup subsets defined by factorization properties dependent on $$\Omega $$ Ω functions

We study combinatorial properties of a class of subsets of semigroups with divisor theory motivated by the study of oscillations of counting functions of sets of algebraic integers with prescribed factorization properties.

Euclidean Quadratic Fields

algebraic field Q(a) is the smallest subfield of the complex numbers which contains a. The algebraic integers I(a) are those elements of Q(a) which are roots of monic polynomials with (ordinary) integer coefficients. Computation in I(a) is in general unlike computation in the integers Z, since usually there is no analogue of the uniqueness of prime factorization. Historically the (false) assumption that prime factorization is unique in every algebraic field proved to be a stumbling block for various distinguished mathematicians, among them Gabriel Lame. (A nice discussion is given by Edwards [6, Chap. 4].) The problem of determining the algebraic fields which do have unique factorization is still not completely solved. However, in certain fields, known as Euclidean fields, it is possible to define an analogue of Euclid's algorithm, and in such cases this guarantees unique factorization. The algebraic fields of degree 2 which have this property are called Euclidean quadratic fields. Work of Davenport and others, culminating in 1952, showed that there are just 21 of them. The well-known book by Hardy and Wright [8] is a standard reference on Euclidean quadratic fields. In 14 of the 21 cases they present proofs that Q(d ) is Euclidean. The reader naturally wonders whether the proofs in the 7 remaining Euclidean cases are difficult. Hardy and Wright also prove that there are no other Euclidean cases with d 0. Indeed, Q(a) = Q(Vd ) in this case. We call this the quadratic field with discriminant d. If the discriminant is negative, the field is complex, and otherwise real. It is convenient to refer to it as Q(Vd ), and to refer to its set of algebraic integers as iG/d ).

Shortest division chains in imaginaryquadratic number fields

Let O d be the set of algebraic integers in an imaginary quadratic number field Q[√ d ], d <0, where d is the discriminant of O d . Consider the Euclidean Algorithm (EA), applied to algebraic integers ξ, η∈ O d . It consists in computing a sequence of remainders ρ 0 =ξ, ρ 1 =η, ρ 2 ,…,ρ n+1 =0, where ρ i+1 =ρ i−1 −γ i ρ i for algebraic integers γ i ∈ O d , i=1,…, n . We show that except for d =−11 the number of divisions to be carried out is always minimized by choosing each γ i such that N(ρ i−1 −γ i ρ i ), the norm of ρ i−1 −γ i ρ i , is minimal. This result has been proven previously in special cases. It also applies to those imaginary quadratic number rings which are not Euclidean; in this case the division chains may be infinite. For d =−7, −8 the methods applied so far must be modified somewhat, and for d =−11 we provide a counterexample and a theorem which partially answers the question, how shortest division chains can be obtained.

On Sets of Algebraic Integers Whose Remaining Conjugates Lie in the Unit Circle

On Sets of Algebraic Integers Whose Remaining Conjugates Lie in the Unit Circle

On sets of algebraic integers whose remaining conjugates lie in the unit circle

On sets of algebraic integers whose remaining conjugates lie in the unit circle

Algebraic integers with two conjugates outside the unit circle

In this paper we consider algebraic integers, θ, such that θ and θ¯ (θ ╪ θ¯) are the only algebraic conjugates of θ that lie outside the unit circle.Let S1 be the set of algebraic integers, ρ, necessarily real, all of whose conjugates, except ρ, satisfy | z &lt; 1.

A Closed Set of Algebraic Integers

A Closed Set of Algebraic Integers