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AbstractLet Fqn be a finite extension of the finite field Fq of degree n, where q is a prime power and n is a positive integer. Suppose a,b∈Fq. In this article we search for pairs (q, n) such that there exists a primitive pair (ξ+ξ−1,f(ξ)) with TrFqn/Fq(ξ)=a and TrFqn/Fq(ξ−1)=b, where ξ is a non-zero element of Fqn and f is any non-exceptional rational function in Fqn(x). [ABSTRACT FROM AUTHOR]

In this paper, we consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. We present the characterization of some known field extensions in terms of polynomial composites. This paper contains the open problem of characterization of ideals in polynomial composites with respect to various field extensions. We also present the full possible characterization of certain field extensions. Moreover, in this paper we show that any finite group is a Galois group of some field extensions and present the inverse Galois problem solved.

The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [On canonical forms. Proc London Math Soc (2). 1920;18:403–410] which was tackled in Fan and Losonczy [Matchings and canonical forms for symmetric tensors. Adv Math. 1996;117(2):228–238]. In this paper, we first discuss unmatchable subsets in abelian groups. Then we formulate and prove linear analogues of results concerning matchings, along with a conjecture that, if true, would extend the primitive subspace theorem. We discuss the dimension m-intersection property for vector spaces and its connection to matching subspaces in a field extension, and we prove the linear version of an intersection property result of certain subsets of a given set. [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]

We study base field extensions of ordinary abelian varieties defined over finite fields using the module theoretic description introduced by Deligne. As applications we give algorithms to determine the minimal field of definition of such a variety and to determine whether two such varieties are twists. [ABSTRACT FROM AUTHOR]

A matching from a finite subset A of an abelian group to another subset B is a bijection f : A → B with the property that a + f (a) never lies in A. A matching is called acyclic if it is uniquely determined by its multiplicity function. Motivated by a question of E. K. Wakeford on canonical forms for symmetric tensors, the study of matchings and acyclic matchings in abelian groups was initiated by C. K. Fan and J. Losonczy in [16,26] , and was later generalized to the context of vector subspaces in a field extension [13,1]. We discuss the acyclic matching and weak acyclic matching properties and we provide results on the existence of acyclic matchings in finite cyclic groups. As for field extensions, we completely classify field extensions with the linear acyclic matching property. The analogy between matchings in abelian groups and in field extensions is highlighted throughout the paper and numerous open questions are presented for further inquiry. [ABSTRACT FROM AUTHOR]

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$ , for which elements $z$ in $\mathbb{L}$ , and $a$ , $b$ in $\mathbb{K}$ , is it possible to write $z$ as a product $xy$ , where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$ ? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets. [ABSTRACT FROM AUTHOR]

Let X, Y be linear spaces over a field K . Assume that f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all x , x i , y , y i ∈ X and with a i , b i ∈ K \ { 0 } , A i , B i ∈ K ( i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all x i , y i ∈ X ( i ∈ { 1 , 2 } ), where C 1 : = A 1 B 1 , C 2 : = A 1 B 2 , C 3 : = A 2 B 1 , C 4 : = A 2 B 2 . We describe the form of solutions and study relations between (∗) and (∗ ∗) . [ABSTRACT FROM AUTHOR]

Let G be a connected simple graph with characteristic polynomial P G (x). The irreducibility of P G (x) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P 2 n + 1 − 2 (n ≥ 1) to certain vertices; union and join K 1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results. [ABSTRACT FROM AUTHOR]

We study the existence of quantum state transfer on non-integral circulant graphs. We find that continuous-time quantum walks on quantum networks based on certain circulant graphs with 2 k k ∈ Z vertices exhibit pretty good state transfer when there is no external dynamic control over the system. We generalize a few previously known results on pretty good state transfer on circulant graphs, and this way we re-discover all integral circulant graphs on 2 k vertices exhibiting perfect state transfer. [ABSTRACT FROM AUTHOR]

We say that a field extension L / F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic 2. [ABSTRACT FROM AUTHOR]

Let A - Z+ and h be positive integer. We say that A is a Bh set if any integer n can be written in at most one-ways as the sum of h elements of A, The fundamental problem is to determine the cardinal maximum of a set Bh contained in the integer interval [1,n] := {1,2,3, . . ., n}. Not many constructions of integer sets Bh are known, among them are Singer [13], Bose-Chowla [3] and Gómez-Trujillo [7]. The Bh set concept can be extended to arbitrary groups. In this article, the generalized constructions on the groups that come from a field are presented and new construction of a set Bh+s in h+1 dimensions is obtained. [ABSTRACT FROM AUTHOR]

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field 𝔽 can be reduced to the following three classifications, for each finite Galois extension 𝕃 of 𝔽 : (1) finite-dimensional central division algebras over 𝕃 , up to isomorphism; (2) twisted group algebras of finite groups over 𝕃 , up to graded-isomorphism; (3) 𝔽 -forms of certain graded matrix algebras with coefficients in Δ ⊗ 𝕃 𝒞 where Δ is as in (1) and 𝒞 is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields. [ABSTRACT FROM AUTHOR]

Abstract Let L / F be a field extension, φ , ψ even-dimensional quadratic forms over F. Assume that the forms φ L and ψ L are similar over L. In the first section we investigate whether it is possible to choose a similarity coefficient from F ⁎. The answer turns out to be positive if the degree of L / F is odd, and negative in general if L / F is a quadratic extension. In the second and third sections we apply the main theorem of the paper to the following problem: Let L (e) / L be a quadratic extension, e ∈ L ⁎ , Φ a form over L. Suppose that dim Φ is even and ind (Φ L (e)) = 1 2 ind (Φ). Does there exist a form Ψ over L such that Ψ L (e) is similar to Φ L (e) , disc (Ψ) = disc (Φ) , and ind (Ψ) = 1 2 ind (Φ) ? We show that the answer is negative in general almost for all conceivable pairs (dim Φ , ind (Φ)). On the other hand, the answer is positive if ind (Φ) = 2 , 4 ≤ dim Φ ≤ 8 , and if ind (Φ) = 4 , dim Φ = 6. In the fourth section we describe the set of similarity coefficients for pairs of 4-dimensional forms. Applying this description, we investigate the CV property with respect to quadratic extensions and pairs of forms (φ , ψ) , where φ and ψ are similar to 2-fold and 1-fold Pfister forms respectively. In the final section we establish general nontriviality of the quotient groups N F (e) / F F (e) ⁎ ∩ F ⁎ 2 Nrd (D F ⁎) F ⁎ 2 N F (e) / F (Nrd (D F (e) ⁎)) and N F (d) / F F (d) ⁎ ∩ N F (e) / F F (e) ⁎ ∩ D (π) F ⁎ 2 N F (d , e) / F (D (π F (d , e))) , where D ∈ Br 2 (F) , and π is a Pfister form over F. [ABSTRACT FROM AUTHOR]

Abstract We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in B α ∈ L [ y ] , where L = Q (α) is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of O ˜ B (N 8) for isolating the real roots of B α , where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes O ˜ B (N 7) if we use Pan's algorithm for isolating the real roots. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of descartes' algorithm. For the former we prove a Boolean complexity bound of O ˜ B (N 12) and for the latter a bound of O ˜ B (N 5). We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials B α k , when α k runs over all the real conjugates of α. We show that we can isolate the real roots of all polynomials in O ˜ B (N 5). Finally, we implemented the algorithms in C as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets. [ABSTRACT FROM AUTHOR]

In this essay, we study various notions of projective space (and other schemes) over F1ℓ, with F1 denoting the field with one element. Our leading motivation is the "Hidden Points Principle," which shows a huge deviation between the set of rational points as closed points defined over F1ℓ and the set of rational points defined as morphisms 𝚂𝚙𝚎𝚌(F1ℓ)↦X. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180–184], schemes of F1ℓ‐type and consider their zeta functions. [ABSTRACT FROM AUTHOR]

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L , respectively. The integral degree of A ⊂ B , denoted by d A (B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between d K (L) and μ A (B), the minimal number of generators of the A -module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous. [ABSTRACT FROM AUTHOR]

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson–Lie algebras: A proof of the Mischenko–Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87–109.)

Secret sharing has been a subject of study since 1979. It is important that a secret key, passwords, information of the map of a secret place or an important formula must be kept secret. The main problem is to divide the secret into pieces instead of storing the whole for a secret sharing. A secret sharing scheme is a way of distributing a secret among a finite set of people such that only some distinguished subsets of these subsets can recover the secret. The collection of these special subsets is called the access structure of the scheme. In this paper, we propose a new approach to construct secret sharing schemes based on field extensions. [ABSTRACT FROM AUTHOR]

Abstract: The homology groups h 1 ( l / k ), h 2 ( l / k ), and h 3 ( l / k ) of the Brauer complex for a triquadratic field extension l = k ( a , b , c ) are studied. In particular, given D ∈ 2 Br ( k ( a , b , c ) / k ), we find equivalent conditions for the image of D in h 2 ( l / k ) to be zero. We consider as well the second divided power operation γ 2 : 2 Br ( l / k ) → H 4 ( k , Z / 2 Z ), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h 2 ⊗ h 1 → H 3, which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F / k such that the cohomology group h 1 ( F ( a , b , c ) / F ) of the Brauer complex contains the images of prescribed elements of k ∗, provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L / F with nontrivial h 3 ( L / F ) are given. As a consequence we show that the homology group h 3 ( L / F ) can be arbitrarily big. [ABSTRACT FROM AUTHOR]

Consider a pair of ordinary elliptic curves E and E ′ defined over the same finite field F q . Suppose they have the same number of F q -rational points, i.e. | E ( F q ) | = | E ′ ( F q ) | . In this paper we characterise for which finite field extensions F q k , k ≥ 1 (if any) the corresponding groups of F q k -rational points are isomorphic, i.e. E ( F q k ) ≅ E ′ ( F q k ) . [ABSTRACT FROM AUTHOR]

Galois Theory, a wonderful part of mathematics with historical roots date back to the solution of cubic and quantic equations in the sixteenth century. However, beside understanding the roots of polynomials, Galois Theory also gave birth to many of the central concepts of modern algebra, including groups and fields. In particular, this theory is further great due to primarily for two factors: first, its surprising link between the group theory and the roots of polynomials and second, the elegance of its presentation. This theory is often descried as one of the most beautiful parts of mathematics. Here I have specially worked on field extensions. To understand the basic concept behind fundamental theory, some necessary Theorems, Lammas and Corollaries are added with suitable examples containing Lattice Diagrams and Tables. In principle, I have presented and solved a number of complex algebraic problems with the help of Galois theory which are designed in the context of various rational and complex numbers. [ABSTRACT FROM AUTHOR]

Let be a field, , , a quartic field extension. We investigate the divided power operation on the group . In particular, we show that any element of is a symbol , where , , and is the quadratic trace form associated with the extension . As an application, we obtain certain results on the Stifel-Whitney maps . [ABSTRACT FROM AUTHOR]