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Given F = p t , a field with p t elements, where p is a prime power and t ≥ 7 is a positive integer. Let n ∈ ℕ and f = f 1 / f 2 is a rational function, where f 1 and f 2 are distinct irreducible polynomials with deg (f 1) + deg (f 2) = n ≤ p t in F [ x ]. We construct a sufficient condition on (p , t) which guarantees primitive pairing (, f ()) exists in F such that Tr p t / p () = a and Tr p t / p (f ()) = b for any prescribed a , b ∈ p . Further, we demonstrate for any positive integer n ≤ p t , such a pair definitely exists for large t. The scenario when n = 2 is handled separately and we verified that such a pair exists for all (p , t) except from possible 71 values of (p , t). A result for the case n = 3 is given as well. [ABSTRACT FROM AUTHOR]

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree. [ABSTRACT FROM AUTHOR]

An extended derivation (endomorphism) of a (restricted) Lie algebra L is an assignment of a derivation (respectively) of L' for any (restricted) Lie morphism f:L\to L', functorial in f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L' to every f; and (b) if L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Given a prime power q and a positive integer n, let 픽 q n {\mathbb{F}_{q^{n}}} represent a finite extension of degree n of the finite field 픽 q {{\mathbb{F}_{q}}} . In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n ≥ 6 {n\geq 6} , q = 3 k {q=3^{k}} , m = 2 {m=2} we establish that there are only 10 possible exceptions. [ABSTRACT FROM AUTHOR]

For a finite field Fqn and a rational function f = f1/f2 ∈ Fqn(x), we present a sufficient condition for the existence of a primitive normal element α ∈ Fqn in such a way f(α) is also primitive in Fqn, where f(x) is a rational function in Fqn(x) of degree sum m (degree sum of f(x) = f1(x)/f2(x) is defined to be the sum of the degrees of f1(x) and f2(x)). Additionally, for rational functions of degree sum 4, we proved that there are only 37 and 16 exceptional values of (q, n) when q = 2k and q = 3k respectively. [ABSTRACT FROM AUTHOR]

In this paper, we show that every quadratic algebraic number lying in a quartic extension L of Q can be written as a quadratic polynomial with rational coefficients in a generator of L over Q . On the other hand, we prove that for each totally real algebraic number β of degree d ≥ 3 , there are infinitely many quadratic extensions L of Q (β) such that β cannot be expressed by a quadratic polynomial with rational coefficients in a generator of L over Q . The same assertion is proved for a large class of cubic algebraic numbers β without assumption that β is totally real. [ABSTRACT FROM AUTHOR]

An element α ∈ F q m is primitive if it is a generator of F q m ∗ . An element β ∈ F q m is normal in F q m over F q if its Galois orbit is a spanning set for F q m as a vector space over F q . If an element is simultaneously primitive and normal, then the element is termed as primitive normal element. In this article, we establish a sufficient condition for the existence of a primitive normal element α ∈ F q m , such that F (α) is also a primitive element of F q m over F q , where F (x) = a x 2 + b x + c d x 2 + e x + f ∈ F q m (x) (with a ≠ 0 , d ≠ 0 ) and F q m is a field of even characteristic. We conclude that every finite field F q m , contains such primitive normal element except for very few q and m. [ABSTRACT FROM AUTHOR]

We prove that for any prime power $q\notin \{3,4,5\}$ , the cubic extension $\mathbb {F}_{q^{3}}$ of the finite field $\mathbb {F}_{q}$ contains a primitive element $\xi $ such that $\xi +\xi ^{-1}$ is also primitive, and $\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi)=a$ for any prescribed $a\in \mathbb {F}_{q}$. This completes the proof of a conjecture of Gupta et al. ['Primitive element pairs with one prescribed trace over a finite field', Finite Fields Appl. 54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree $n\ge 3$. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let F q n be a finite field with q n elements, and let m 1 and m 2 be positive integers. Given polynomials f 1 (x) , f 2 (x) ∈ F q n [ x ] with deg (f i (x)) ≤ m i , for i = 1 , 2 , and such that the rational function f 1 (x) / f 2 (x) satisfies certain conditions which we define, we present a sufficient condition for the existence of a primitive element α ∈ F q n , normal over F q , such that f 1 (α) / f 2 (α) is also primitive. [ABSTRACT FROM AUTHOR]

Irregular geographic spatio-temporal-field data have been rapidly accumulating; however, data organizations and operations for different irregular types are often segregated, leading to systematic drawbacks, such as interface expansion difficulty and high coupling codes in GIS implementations. The paper proposes a unified approach to organizing and operating irregular geographic spatio-temporal-field data. The proposed approach has two components, namely 'concepts and definitions', and 'logical model'. The first component introduces the concept of primitive elements, which are formal sets of data points, to serve as the smallest building blocks in the data organization. We define the corresponding primitive elements for three prevalent irregularity types (including sparse, imbalanced, and heterogeneous). The second component utilizes object-oriented programming to support the implementation of various operators. Additionally, we develop the layered architecture to decouple data organization, operation, and visualization to assure low coupling among layers. For demonstrations, we conduct case studies to show the effectiveness of our approach. Additionally, we conduct experiments to new irregularity types and illustrate the flexibility and scalability of our approach. Comparisons with classic tensor methods and spatio-temporal analysis methods show that our approach has more comprehensive supports for different data types. [ABSTRACT FROM AUTHOR]

Let α and β be two algebraic numbers, F = Q (α , β) and d = [ F : Q ] ≥ 2 . By the primitive element theorem, for all but finitely many rational numbers r we have F = Q (α + r β) . A straightforward argument implies that the number of exceptional r, namely, those r ∈ Q for which Q (α + r β) is a proper subfield of F, is at most (d - 1) 2 . We show that the number of exceptional r is at most d. On the other hand, we give an example showing the number of exceptional r can be greater than ( log d log log d ) 2 for infinitely many d ∈ N . [ABSTRACT FROM AUTHOR]

We introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of multi-complexes which is defined as the algebra which has a formal basis C of all isomorphism types of multi-complexes, and multiplication is to take the disjoint union. This is a Hopf algebra with an operation encoding the disassembly information for such objects and extends the Hopf algebra of graphs. In our main result, we explicitly describe here the structure of this Hopf algebra of multi-complexes H. We find an explicit basis B of the space of primitives, which is of combinatorial relevance: it is such that each multi-complex is a polynomial with non-negative integer coefficients of the elements of B , and each b ∈ B is a polynomial with integer coefficients in C . Using this, we find the grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of sub-multi-complexes in a multi-complex. We also explicitly illustrate how our results specialize to the graph Hopf algebra, and observe how they specialize to results in all of the above-mentioned particular cases. We also investigate applications of these results to the graph reconstruction conjectures and rederive some results in the literature on these questions. [ABSTRACT FROM AUTHOR]

For q = 3 r (r ∈ ℕ), denote by q the finite field of order q and for a positive integer m ≥ 2 , let q m be its extension field of degree m. We establish a sufficient condition for existence of a primitive normal element α such that f (α) is a primitive element, where f (x) = a x 2 + b x + c , with a , b , c ∈ q m satisfying b 2 ≠ a c in q m . [ABSTRACT FROM AUTHOR]

Let q be an even prime power and m ≥ 2 an integer. By q , we denote the finite field of order q and by q m its extension of degree m. In this paper, we investigate the existence of a primitive normal pair (α , f (α)) , with f (x) = a x 2 + b x + c d x + e ∈ q m (x) where the rank of the matrix F = a b c 0 d e ∈ M 2 × 3 ( q m ) is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for 1 1 0 0 1 0 if q = 2 and m is odd, and then we provide an explicit small list of possible and genuine exceptional pairs (q , m). [ABSTRACT FROM AUTHOR]

For a prime p and a positive integer k , let q = p k and F q n be the extension field of F q. We derive a sufficient condition for the existence of a primitive element α in F q n such that α 3 − α + 1 is also a primitive element of F q n , a sufficient condition for the existence of a primitive normal element α in F q n over F q such that α 3 − α + 1 is a primitive element of F q n , and a sufficient condition for the existence of a primitive normal element α in F q n over F q such that α 3 − α + 1 is also a primitive normal element of F q n over F q. [ABSTRACT FROM AUTHOR]

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^{n}-1$. We say that the extension $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements property if, for every $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that $\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$ , there exists some $x\in \mathbb{F}_{q}$ such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$. We prove that, for sufficiently large prime powers $q$ , $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements. We also discuss the (weaker) translate property for extensions. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Given m , n , q ∈ N such that q is a prime power and m ≥ 3 , a ∈ F q , we establish a sufficient condition for the existence of a primitive pair (α , f (α)) in F q m such that α is normal over F q and Tr F q m / F q (α - 1) = a , where f (x) ∈ F q m (x) is a rational function of degree sum n. Further, when n = 2 and q = 5 k for some k ∈ N , such a pair definitely exists for all (q, m) apart from at most 20 choices. [ABSTRACT FROM AUTHOR]

In this paper, we give a largely self-contained proof that the quartic extension q 4 of the finite field q contains a primitive element α such that the element α + α − 1 is also a primitive element of q 4 , and Tr q 4 | q (α) = a for any prescribed a ∈ q . The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl.54 (2018) 1–14.]. [ABSTRACT FROM AUTHOR]

We exploit structure in polynomial system solving by considering polynomials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka generalized Sylvester matrices. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex. The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of Canny, Kaltofen, and Yagati Canny et al. (1989) to the multihomogeneous case. We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success 1 − 1 / 2 ϱ , for a given ϱ ≥ 1 , and bit complexity O ˜ B (n 2 D 4 + ϵ (n N + 1 + τ) + n D 2 + ϵ ϱ (D + ϱ)) for any ϵ > 0 , where n is the number of variables, D equals the multilinear Bézout bound, N is the number of variable subsets, and τ is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input. [ABSTRACT FROM AUTHOR]

Given a Hopf algebra with distinguished bases indexed by combinatorial objects, along with a primitive generating set for this algebra, it is natural to consider how we can combinatorially interpret the primitivity of the elements in this generating set, by expanding these generators in terms of the distinguished bases of this algebra and then applying the comultiplication operation to these expansions and constructing sign-reversing involutions that determine the resultant cancellations that we obtain. In this article, we explore this idea, as applied to the power sum generators of the algebra of symmetric functions. [ABSTRACT FROM AUTHOR]

A signed graph is a pair (G , τ) of a graph G and its sign τ , where a sign τ is a function from { (e , v) ∣ e ∈ E (G) , v ∈ V (G) , v ∈ e } to { 1 , - 1 } . Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal I (G , τ) associated with a signed graph (G , τ) , and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of I (G , τ) and then focus on the complete intersection property. More precisely, we find a complete list of graphs G such that I (G , τ) is a complete intersection for every sign τ . [ABSTRACT FROM AUTHOR]

Let F = F q m , m ≥ 7 , n a positive integer, and f = p 1 / p 2 with p1, p2 co-prime irreducible polynomials in F [ x ] and deg (p 1) + deg (p 2) = n. We obtain a sufficient condition on (q, m), which guarantees, for any prescribed a, b in E = F q , the existence of primitive pair (α , f (α)) in F such that Tr F / E (α) = a and Tr F / E (α − 1) = b. Further, for every positive integer n, such a pair definitely exists for large enough m. The case n = 2 is dealt separately and proved that such a pair exists for all (q, m) apart from at most 64 choices. [ABSTRACT FROM AUTHOR]

Given a prime power q and an integer n ≥ 2 , we establish a sufficient condition for the existence of a primitive pair (α , f (α)) where α ∈ F q and f (x) ∈ F q (x) is a rational function of degree sum n. (Here f = f 1 / f 2 , where f 1 , f 2 are coprime polynomials of degree n 1 , n 2 , respectively, and the sum of their degrees n 1 + n 2 = n.) For any n , such a pair is guaranteed to exist for sufficiently large q. Indeed, when n = 2 , such a pair definitely does not exist only for 28 values of q and possibly (but unlikely) only for at most 3911 other values of q. [ABSTRACT FROM AUTHOR]

We determine lower bounds for the number of primitive normal elements in a four-dimensional extension E over a Galois field F = GF (q) . Our approach is based on viewing E as the three-dimensional projective space Γ = PG (3 , q) . In any of the three cases, whether q is even, or q ≡ 3 mod 4 , or q ≡ 1 mod 4 , we use a decomposition of the multiplicative group of E in order to determine a (canonical) partition of the point set of Γ into q + 1 ovoids. The points of Γ are distinguished into primitive and non-primitive ones, and an ovoid is called primitive if it contains at least one primitive point. The bounds are derived by studying the intersections of the primitive ovoids with the configuration of those points of Γ which do not give rise to normal elements of E / F. Given that q 2 + 1 is a prime number when q is even, or that 1 2 (q 2 + 1) is a prime number when q is odd, we actually achieve the exact number of all primitive normal elements for the quartic extension over F. [ABSTRACT FROM AUTHOR]

In this article we present an algorithm to determine the smallest primitive element for Z* where p is a safe prime. We also address the case of primes of the form p = 2kq + 1 for an odd prime q. [ABSTRACT FROM AUTHOR]

Let r , n > 1 be integers and q be any prime power q such that r   |   q n − 1. We say that the extension 𝔽 q n / 𝔽 q possesses the line property for r -primitive elements if, for every α , 𝜃 ∈ 𝔽 q n ∗ , such that 𝔽 q n = 𝔽 q (𝜃) , there exists some x ∈ 𝔽 q , such that α (𝜃 + x) has multiplicative order (q n − 1) / r. Likewise, if, in the above definition, α is restricted to the value 1 , we say that 𝔽 q n / 𝔽 q possesses the translate property. In this paper, we take r = n = 2 (so that necessarily q is odd) and prove that 𝔽 q 2 / 𝔽 q possesses the translate property for 2-primitive elements unless q ∈ { 5 , 7 , 1 1 , 1 3 , 3 1 , 4 1 }. With some additional theoretical and computational effort, we show also that 𝔽 q 2 / 𝔽 q possesses the line property for 2-primitive elements unless q ∈ { 3 , 5 , 7 , 9 , 1 1 , 1 3 , 3 1 , 4 1 }. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements. Quadratic residues in number theory and finite fields is an important theory that has many applications in various aspects. The main problem of quadratic residues is to find the solution of the equation x2 = a, given an element a. It is interesting to find the solutions of x3 = a in F q. If the solutions exist for a we say that a is a cubic residue of F q and x is a cube root of a in F q. In this paper we examine the solubility of x3 = a in general finite fields. Here, we give some results about the cube roots of cubic residue, and we propose an algorithm to find the cube roots using primitive elements. [ABSTRACT FROM AUTHOR]

Let F q be the finite field of characteristic p with q elements and F q n its extension of degree n. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension F q n / F q for any q and n. It is known that the conjecture holds for n ≤ q . In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for q ≤ n ≤ O (q ϵ) , where ϵ = 2 for the asymptotic results and ϵ = 1.25 for the effective ones. For n even we need to assume that q - 1 ∤ n . [ABSTRACT FROM AUTHOR]

It’s important to find an orthogonal set of transmitting waveforms of MIMO sonar, and the orthogonality of signal set has a direct affect on the detection efficiency of MIMO sonar. The frequency-hopped signal based on a Costas sequence has a nearly ideal range-Doppler ambiguity function, however, the different frequency-hopped signals constructed by different primitive elements of a finite field are not always orthogonal. A search method for primitive elements is proposed based on the finite field theory, and an optimal orthogonal set of transmitting waveforms is designed based on the primitive elements of safe primes. The simulated results show that a set of transmitting waveforms made from a safe prime, which is less than 1 000, has not only the best orthogonality but also the maximum number of signals. [ABSTRACT FROM AUTHOR]

The Oeljeklaus–Toma (OT-)manifolds are compact, complex, non-Kähler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field K and a torsion-free subgroup U in the group of units of the ring of integers of K, with rank of U equal to the number of real embeddings of K. OT-manifolds are equipped with a torsion-free flat affine connection preserving the complex structure (this structure is known as "flat affine structure"). We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all elements in U \ { 1 } are primitive in K, then X contains no proper analytic subvarieties. [ABSTRACT FROM AUTHOR]

In this article, we establish a sufficient condition for the existence of a primitive element such that for any matrix of rank 2, the element is a primitive element of and , for any prescribed , where for some positive integer k. We establish existence of such elements for all pairs except for finitely many pairs. [ABSTRACT FROM AUTHOR]