On numbers which are differences of two conjugates of an algebraic integer

Additive Hilbert's Theorem 90 in the ring of algebraic integers

Computational complexity of quantifier-free negationless theory of field of rational numbers

In this paper, algorithms for computing the minimal polynomial and the common minimal polynomial of resultant matrices over any field are presented by means of the approach for the Grobner basis of the ideal in the polynomial ring, respectively, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with resultant blocks over any field is given, which can be realized by CoCoA 4.0, an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. We get examples showing the effectiveness of the algorithms.

A Gröbner Free Alternative for Polynomial System Solving

The problem of the periodicity of functional continued fractions of elements of a hyperelliptic field is closely related to the problem of finding and constructing fundamental S-units of a hyperelliptic field and the torsion problem in the Jacobian of the corresponding hyperelliptic curve. For elliptic curves over a field of rational numbers, the torsion problem was solved by B. Mazur in 1978. For hyperelliptic curves of genus 2 and higher over the field of rational numbers, the above three problems remain open. The theory of functional continued fractions has become a powerful arithmetic tool for studying these problems. In addition, tasks arising in the theory of functional continued fractions have their own interest. Sometimes these tasks have analogues in the numerical case, but tasks that are significantly different from the numerical case are especially interesting. One such problem is the problem of estimating from above the lengths of periods of functional continued fractions of elements of a hyperelliptic field over a field of rational numbers. In this article, we find upper bounds on the period lengths for key elements of a hyperelliptic field over a field of rational numbers. In the case when the hyperelliptic field is defined by an odd degree polynomial, the period length of the elements under consideration is either infinite or does not exceed twice the degree of the fundamental S-unit. A more interesting and complicated case is when a hyperelliptic field is defined by a polynomial of even degree. In 2019, V. P. Platonov and G. V. Fedorov for hyperelliptic fields $$L = \mathbb{Q}(x)(\sqrt{f}), \deg f = 2g + 2,$$ found the exact interval values $$s \in \mathbb{Z}$$ such that continued fractions of elements of the form $$\sqrt{f}/h^s \in L \setminus \mathbb{Q}(x)$$ are periodic. Using this result in this article, we find exact upper bounds on the period lengths of functional continued fractions of elements of a hyperelliptic field over a field of rational numbers, depending only on the genus of the hyperelliptic field and the order of the torsion group of the Jacobian of the corresponding hyperelliptic curve.

On prime divisors of the index of an algebraic integer

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.

In this thesis, we study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. DAVENPORT and W.M. SCHMIDT in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality requires to study the dual problem of approximation to successive powers of these numbers by rational numbers with the same denominators. Dirichlet's Box Principle provides estimates for such approximations but one can do better. In this thesis we establish constraints on how much better one can do when dealing with the numbers and their squares. We also construct examples showing that at least in some instances these constraints are optimal. Going back to the original problem, we obtain estimates for simultaneous approximation to real and p-adic numbers by roots of integer polynomials of degree 3 or 4 with fixed coefficients in degree ≥ 3. In the case of a single real number (and no p-adic numbers), we extend work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace.

Minimal Polynomials and Distinctness of Kloosterman Sums

Let \(\alpha \) be an algebraic number of degree \(d\) with minimal polynomial \(F \in \mathbb {Z}[X]\), and let \(\mathbb {Z}[\alpha ]\) be the ring generated by \(\alpha \) over \(\mathbb {Z}\). We are interested whether a given number \(\beta \in \mathbb {Q}(\alpha )\) belongs to the ring \(\mathbb {Z}[\alpha ]\) or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number \(r/t \in \mathbb {Q}\), where \(r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1\), belongs to the ring \(\mathbb {Z}[\alpha ]\) if and only if the square-free part of its denominator \(t\) divides all the coefficients of the minimal polynomial \(F \in \mathbb {Z}[X]\) except for the constant coefficient \(F(0)\) that must be relatively prime to \(t\), namely \(\gcd (F(0),t)=1\). We also study the question when the equality \(\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']\) for algebraic numbers \(\alpha , \alpha '\) conjugates over \(\mathbb {Q}\) holds. In particular, it is shown that for each \(d \in \mathbb {N}\), there are conjugate algebraic numbers \(\alpha , \alpha '\) of degree \(d\) satisfying \(\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')\) and \(\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']\). The question concerning the equality \(\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']\) is answered completely for conjugate quadratic pairs \(\alpha ,\alpha '\) and also for conjugate pairs \(\alpha , \alpha '\) of cubic algebraic integers.

Circulant matrices have important applications in solving various differential equations. The level-kscaled factor circulant matrix over any field is introduced. Algorithms for finding the minimal polynomial of this kind of matrices over any field are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring. And two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for computing the inverse of partitioned matrix with level-kscaled factor circulant matrix blocks over any field is given by using the Schur complement, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo prime number.

this article. This deficiency will be described at some length later in this review, but given the present politically charged climate in mathematics education, let me hasten to add that teachers themselves are not the sole cause of this deficiency. Inadequate mathematics instruction in universities and low salaries in the teaching profession also have a lot to do with it

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

Radix representations of quadratic fields

Units An algebraic integer e is said to be a unit if 1/e is an algebraic integer. This…is equivalent to the condition N e = ± 1. Indeed the conjugates of an algebraic integer are again algebraic integers, whence, if e is a unit, then N e and 1/ N e are rational integers and so ±1. Conversely if N e = ±1 then 1/e = ± N e/e which is clearly an algebraic integer. The set of all units form a group U under multiplication and the set of units in a number field K form a subgroup U K . Further, we see that if [α], [β] are principal ideals in K then we have [α] = [β] if and only if α/β ∈ U K . In general, we say that non-zero algebraic numbers α, β are associated if α/β ∈ U . The units in ℚ are plainly ±1 whence they are all the roots of unity in ℚ, that is, the solutions of an equation x l = 1 for some positive integer l . The units of the quadratic field K = ℚ(√ d ), where d ≠ 1 is a square-free integer, were discussed in Section 7.3. It was shown there that for the imaginary quadratic field K with d x 2 − 1 for D x 4 − 1 for D = −4 and of x 6 − 1 for D = −3 where D denotes the discriminant of the field, that is, D = 4 d for d ≡ 2, 3 (mod4) and D = d for d ≡ 1 (mod4).