Algebraic Number Rings Research Articles

Frieze patterns over algebraic numbers

AbstractConway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field where . We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases all non‐zero frieze patterns over the rings of integers for have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns.

Finer factorization characterizations of class number 2

In [4], class number 2 is characterized among algebraic number rings using basic factorization tools. In this note, we extend these characterizations using finer factorization invariants.

Quotients of primes in an algebraic number ring

Quotients of primes in an algebraic number ring

Finding Elements With Given Factorization Lengths and Multiplicities

Many algebraic number rings exhibit nonunique factorization of elements into irreducibles. Not only can the irreducibles in the factorizations be different, but the number of irreducibles in the factorizations can also vary. A basic question then is: Which sets can occur as the set of factorization lengths of an element? Moreover, how often can each factorization length occur? While these questions are most pertinent in algebraic number rings, their pertinence extends to Dedekind domains and a broader class of structures called Krull monoids. Surprisingly, for a large subclass of Krull monoids, Kainrath was able to resolve completely the question of which length sets and length multiplicities can be realized. In this article, we explain the context of Kainrath's theorem and give a constructive proof for an important case, namely Krull monoids with infinite nontorsion class group. We also construct length sets in a case not covered by Kainrath's theorem to illustrate the difficulty of the general problem.

On certain pure sextic fields related to a problem of Hasse

An algebraic number ring is monogenic, or one-generated, if it has the form [Formula: see text] for a single algebraic integer [Formula: see text]. It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if [Formula: see text] is a square-free rational integer, [Formula: see text] and [Formula: see text], then the pure sextic field [Formula: see text] is not monogenic. Our results are illustrated by examples.

On integral absolutely irreducible representations of finite groups

We consider the arithmetic background of integral representations of finite groups over [Formula: see text]-adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite [Formula: see text]-groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed.

Rings of algebraic numbers in infinite extensions of $${\mathbb {Q}}$$ and elliptic curves retaining their rank

We show that elliptic curves whose Mordell–Weil groups are finitely generated over some infinite extensions of \({\mathbb {Q}}\) , can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers

Using Poonen’s version of the “weak vertical method” we produce new examples of “large” and “small” rings of algebraic numbers (including rings of integers) where $\mathbb {Z}$ and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture on the topology of rational points does not hold.

Higher composition laws I: A new view on Gauss composition, and quadratic generalizations

Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow. Even today, two centuries later, this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders. It is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields. This article forms the first of a series of four articles in which our aim is precisely to develop such “higher composition laws”. In fact, we show that Gauss’s law of composition is only one of at least fourteen composition laws of its kind which yield information on number rings and their class groups. In this paper, we begin by deriving a general law of composition on 2×2×2 cubes of integers, from which we are able to obtain Gauss’s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves. We also obtain from this composition law on 2× 2 × 2 cubes four further new laws of composition. These laws of composition are defined on 1) binary cubic forms, 2) pairs of binary quadratic forms, 3) pairs of quaternary alternating 2-forms, and 4) senary (six-variable) alternating 3-forms. More precisely, Gauss’s theorem states that the set of SL2(Z)-equivalence classes of primitive binary quadratic forms of a given discriminant D has an inherent group structure. The five other spaces of forms mentioned above (including the space of 2 × 2 × 2 cubes) also possess natural actions by special linear groups over Z and certain products thereof. We prove that, just like Gauss’s space of binary quadratic forms, each of these group actions has the following remarkable properties. First, each of these six spaces possesses only a single polynomial invariant for the corresponding group action, which we call the discriminant. This discriminant invariant is found to take only values that

Warfield Domains: Module Theory from Linear Algebra to Commutative Algebra Through Abelian Groups

In this survey paper we illustrate the generalizations to modules over integral domains of the basic result of linear algebra stating that every finite dimensional vector space is canonically isomorphic to its bidual. These generalizations culminate in the discovery of Warfield domains. We present the characterization of Warfield domains in terms of properties of their overrings, as well as their intrinsic characterizations in the classical cases of Noetherian and integrally closed domains, and in the general case. Finally, the development of the results on direct sum decompositions of torsionless modules from the classical case of Dedekind domains up to Warfield domains is sketched; this subject was originated by the 1912 result of Steinitz on finite direct sums of ideals in algebraic number rings.

AN ARITHMETICAL CHARACTERIZATION OF FINITE ELEMENTARY 2-GROUPS

Beginning with Carlitz's well known characterization of algebraic number rings with class number two, arithmetical characterizations of divisor class groups have been a topic of interest in the literature. In this note we develop a characterization of Dedekind domains with finite elementary 2-group class group. The characterization is in terms of the asymptotic behavior of the number of distinct factorizations of powers of an irreducible element.

Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator

Let K be a number field. Let W be a set of non-archimedean primes of K, let O K , W ={x∈K∣ord p x≥0∀p∉W}. Then if K is a totally real non-trivial cyclic extension of ℚ, there exists an infinite set W of finite primes of K such that ℤ and the ring of algebraic integers of K have a Diophantine definition over O K , W . (Thus, the Diophantine problem of O K , W is undecidable.)

Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of [formula omitted

This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of Q where Hilbert's Tenth Problem is undecidable.

On the Lengths of Factorizations of Elements in an Algebraic Number Ring

On the Lengths of Factorizations of Elements in an Algebraic Number Ring

On the Lengths of Factorizations of Elements in an Algebraic Number Ring

Let D be an integral domain in which each nonzero nonunit can be written as a finite product of irreducible elements from D (such a domain is called atomic). For any positive integer n, let V ( n) be the set of all integers m for which there exists irreducible elements α 1, ..., α n , β 1, ..., β m of D such that α 1 · · · α n = β 1 · · · β m . We then set Φ(n) = | V(n)| . In this paper, we consider this function and its asymptotic behavior for a large class of Dedekind domains including rings of integers of algebraic number fields). In particular we prove the following. THEOREM. Let D be a Dedekind domain with finite class group G such that every ideal class contains at least one prime ideal; let D( G) be the Davenport constant of G ( see [4]), then,[formula].

A Diophantine definition of rational integers over some rings of algebraic numbers.

The author considers the rings of algebraic numbers integral at all but finitely many primes in the number fields, where it has been previously shown that Hubert's Tenth Problem is undecidable in the rings of algebraic integers, and proves that the problem is still undecidable in the bigger rings by constructing a diophantine definition of rational integers there.

Undecidable Existential Problems for Addition and Divisibility in Algebraic Number Rings

Undecidable Existential Problems for Addition and Divisibility in Algebraic Number Rings

Undecidable existential problems for addition and divisibility in algebraic number rings

Existential formulas involving addition and divisibility are shown to be undecidable in the ring of integers of a real quadratic extension of the rationals. A weaker result is proved for extensions of higher degree.

Undecidable Existential Problems for Addition and Divisibility in Algebraic Number Rings. II

It is shown that for all algebraic number rings, except imaginary quadratic ones, the problem of deciding existential formulas involving only addition and the divisibility predicate is equivalent to the full diophantine problem for these rings.

Undecidable existential problems for addition and divisibility in algebraic number rings. II

It is shown that for all algebraic number rings, except imaginary quadratic ones, the problem of deciding existential formulas involving only addition and the divisibility predicate is equivalent to the full diophantine problem for these rings.