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).

Unit group structure of the quotient ring of a quadratic ring

Let [Formula: see text] be the rational filed. For a square-free integer [Formula: see text] with [Formula: see text], we denote by [Formula: see text] the quadratic field. Let [Formula: see text] be the ring of algebraic integers of [Formula: see text]. In this paper, we completely determine the unit group of the quotient ring [Formula: see text] of [Formula: see text] for an arbitrary prime [Formula: see text] in [Formula: see text], where [Formula: see text] has the unique factorization property, and [Formula: see text] is a rational integer.

Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

Integral trace form of extensions of degree pq

In this work, we present the integral trace form [Formula: see text] of a cyclic extension [Formula: see text] with degree [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct odd primes, the conductor of [Formula: see text] is a square free integer, and [Formula: see text] belongs to the ring of algebraic integers [Formula: see text] of [Formula: see text]. The integral trace form of [Formula: see text] allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of [Formula: see text].

Constructions of Dense Lattices over Number Fields

In this work, we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2,3,4,5,6,8 and 12, which are rotated versions of the lattices Lambda_n, for n =2,3,4,5,6,8 and K_12. These algebraic lattices are constructed through canonical homomorphism via Z-modules of the ring of algebraic integers of a number field.

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.

Integral bases of pure fields with square-free parameter

Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases. In this paper we explicitly give an integral basis of the field , where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of is repeating periodically in m with period length depending on n.

Diophantine problems in solvable groups

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].

On the number of incongruent solutions to a quadratic congruence over algebraic integers

Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed that [Formula: see text] is unramified over [Formula: see text]. Moreover, some interesting identities for exponential sums are proved.