Algebraic Number Field Research Articles

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).

Two-parameter identities for divisor sums in algebraic number fields

In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an “extra” parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with fK(n), the number of integral ideals of norm n in an algebraic number field K. In this paper we establish Riesz sum identities containing an “extra” parameter and involving fK(n), or divisor functions associated with K. Upper bounds for the sums as the upper index tends to infinity are also established.

On nonmonogenic number fields defined by

AbstractLet q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.

Some Properties of Euler’s Function and of the Function τ and Their Generalizations in Algebraic Number Fields

In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields.

On the modular forms of weight 1/2 over algebraic number fields

Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of modular forms of weight 1/2 over imaginary quadratic fields.In this paper, we determine an explicit basis of the space of modular forms of weight 1/2, level c and character ψ over algebraic number fields. We prove our assertion using their arguments and Shimura's transformation formula of theta series over algebraic number fields.

ELLIPTIC CURVES AND -ADIC ELLIPTIC TRANSCENDENCE

AbstractWe prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

Explicit integral basis of pure sextic fields

Let K=ℚ(𝜃) be an algebraic number field with 𝜃 satisfying an irreducible polynomial x6−m having integer coefficient m. We explicitly provide an integral basis of K along with some examples. As an application, we show that when m is a squarefree integer such that m≡2,3 mod4 and m≢±1 mod9, then K has a power integral basis.

Function Theories in Cayley-Dickson Algebras and Number Theory

In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions.An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example: Cayley-Dickson algebras admit the construction of direct analogues of so-called CM-lattices, in particular, lattices that are closed under multiplication.Canonical examples are lattices with components from the algebraic number fields mathbb{Q}{[sqrt{m1}, ldots sqrt{mk}]}. Note that the multiplication of two non-integer lattice paravectors does not give anymore a lattice paravector in the Clifford algebra. In this paper we exploit the tools of octonionic function theory to set up an algebraic relation between different octonionic generalized elliptic functions which give rise to octonionic elliptic curves. We present explicit formulas for the trace of the octonionic CM-division values.

A note on values of the Dedekind zeta-function at odd positive integers

For an algebraic number field [Formula: see text], let [Formula: see text] be the associated Dedekind zeta-function. It is conjectured that [Formula: see text] is transcendental for any positive integer [Formula: see text]. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when [Formula: see text] is a totally real number field, [Formula: see text] is an algebraic multiple of [Formula: see text] and hence, is transcendental. If [Formula: see text] is not totally real, the question of whether [Formula: see text] is irrational or not remains open. In this paper, we prove that for a fixed integer [Formula: see text], at most one of [Formula: see text] is rational, as [Formula: see text] varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields.

On clean, weakly clean and feebly clean commutative group rings

A ring [Formula: see text] is said to be clean if each element of [Formula: see text] can be written as the sum of a unit and an idempotent. [Formula: see text] is said to be weakly clean if each element of [Formula: see text] is either a sum or a difference of a unit and an idempotent, and [Formula: see text] is said to be feebly clean if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text] are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] are distinct primes. In this paper, we consider a more general setting. Let [Formula: see text] be an algebraic number field, [Formula: see text] its ring of integers, [Formula: see text] a nonzero prime ideal and [Formula: see text] the localization of [Formula: see text] at [Formula: see text]. We investigate when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when [Formula: see text] is a cyclotomic field or [Formula: see text] is a quadratic field.

On multiple rational trigonometric sums over a field of algebraic numbers

On multiple rational trigonometric sums over a field of algebraic numbers

Infinite linear and algebraic independence of values of F-series at polyadic Liouvillea points

This paper proves infinite linear and algebraic independence of the values of 𝐹-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. 𝐹- series have form 𝑓𝑛 =Σ∞𝑛=0 𝑎𝑛𝑛!𝑧𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties. These series converge in the field Q𝑝 of 𝑝-adic numbers and their algebraic extensions K𝑣. Polyadic number is a series of the form Σ∞𝑛=0 𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞|< 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼−𝐴|𝑝 < 𝐴−𝐷. Infinite linear (algebraic) independence means that for any nonzero linear form (any nonzero polynomial) there are infinitely many primes 𝑝 and valuations 𝑣 extending 𝑝-adic valuation to an algebraic number field K with the following property: the result of substitution in the considered linear form (polynomial) of the values of 𝐹 - of series instead of variables is a nonzero element of the field. Previously, only the existence of at least one prime number 𝑝 with the properties listed above was proved.

Entire functions with undecidable arithmetic properties

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f(z)=∑k=0∞akzk where the coefficients ak∈K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f(α) is algebraic or transcendental for non-zero algebraic arguments α. For example, if f(z) is a transcendental Mahler function, then under generic conditions f(α) is transcendental for all non-zero algebraic numbers with |α|<1. Also, if f(z) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f(n)(α) for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions f(z), g(z), and h(z) with computable rational coefficients for which no algorithms exist that determine if f(n)∈Q, g(n)(1)∈Q, or ∫01h(z)zndz∈Q for integral n≥0. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9].

NOTES ON THE K-RATIONAL DISTANCE PROBLEM

AbstractLet K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every $n\geq 2$ in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square.

On primes dividing the index of a quadrinomial

Let ℤK denote the ring of algebraic integers of an algebraic number field K=ℚ(𝜃) where the algebraic integer 𝜃 is a root of an irreducible quadrinomial f(x)=xn+axn−1+bxn−2+c belonging to ℤ[x] with a2=4b. We give necessary and sufficient conditions involving only a,b,c,n for a prime p to divide the index of the subgroup ℤ[𝜃] in ℤK. As a consequence, we obtain necessary and sufficient conditions for ℤK to be equal to ℤ[𝜃]. Moreover, when ℤK≠ℤ[𝜃], we provide an explicit formula for the index [ℤK:ℤ[𝜃]] in some cases.

On Greenberg’s generalized conjecture for imaginary quartic fields

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

On the Finiteness of the Number of Expansions into a Continued Fraction of $$\sqrt f $$ for Cubic Polynomials over Algebraic Number Fields

We obtain a complete description of cubic polynomials f over algebraic number fields $$\mathbb{K}$$ of degree $$3$$ over $$\mathbb{Q}$$ for which the continued fraction expansion of $$\sqrt f $$ in the field of formal power series $$\mathbb{K}((x))$$ is periodic. We also prove a finiteness theorem for cubic polynomials $$f \in K[x]$$ with a periodic expansion of $$\sqrt f $$ for extensions of $$\mathbb{Q}$$ of degree at most 6. Additionally, we give a complete description of such polynomials f over an arbitrary field corresponding to elliptic fields with a torsion point of order $$N \geqslant 30$$ .

Hilbert modular forms and codes over [formula omitted

Let p be an odd prime and consider the finite field Fp2. Given a linear code C⊂Fp2n, we use algebraic number theory to construct an associated lattice ΛC⊂OLn for L an algebraic number field and OL the ring of integers of L. We attach a theta series θΛC to the lattice ΛC and prove a relation between θΛC and the complete weight enumerator evaluated on weight one theta series.

Some generalizations of the functions [formula omitted] and [formula omitted] in algebraic number fields

In this paper, we generalize the arithmetic functions τ and τe in algebraic number fields and we find some properties of these functions.

Some Properties of Extended Euler’s Function and Extended Dedekind’s Function

In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.