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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]

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]

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]

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]

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]

We present a Graphical Processing Unit (GPU) accelerated sharp interface Immersed Boundary (IB) method that can be applied to versatile geometries on a staggered Cartesian grid. The current IB solver predicts the flow around arbitrary surfaces of both finite and negligible thicknesses with improved accuracy near the sharp edges. The proposed methodology first uses a modified signed distance algorithm to track the complex geometries on the structured Cartesian grid accurately. Afterwards, we impose the boundary conditions by reconstructing the flow variables on the near boundary nodes in both fluid and solid domains. We have also shown a reduction of Spurious Force Oscillations (SFOs) near the moving boundaries with reduced divergence error. The accuracy of the present solver is demonstrated at low Reynolds numbers over different stationary and moving rigid geometries associated with sharp edges pertaining to several engineering applications. We have discussed the steps for GPU optimisation of the present solver. Our implementation ensures the concurrent execution of threads for the field extension-based velocity and pressure reconstruction algorithm on a GPU. More than 100x speedup is obtained on NVIDIA V100 GPU for the three-dimensional oscillating sphere simulation. It is observed that the speedup is higher for larger mesh sizes. The computational performance over both the multi-core Control Processing Units (CPUs) and NVIDIA GPUs (V100 and A100) using OpenACC is also provided for the insect flow simulation. • A field extension based sharp IB method for bluff bodies and thin surfaces. • Improved accuracy near sharp edges using multi-directional forcing. • A surface tracking algorithm based on modified signed distance approach. • GPU optimisation strategy with reduced warp divergence. [ABSTRACT FROM AUTHOR]

We consider differential spatial modulation (DSM) operating in a block fading environment and propose sparse unitary dispersion matrices (DMs) using algebraic field extensions. The proposed DM sets are capable of exploiting full transmit diversity and, in contrast to the existing schemes, can be constructed for systems having an arbitrary number of transmit antennas. More specifically, two schemes are proposed: 1) field-extension-based DSM (FE-DSM), where only a single conventional symbol is transmitted per space–time block; and 2) FE-DSM striking a diversity–rate tradeoff (FE-DSM-DR), where multiple symbols are transmitted in each space–time block at the cost of a reduced transmit diversity gain. Furthermore, the FE-DSM scheme is analytically shown to achieve full transmit diversity, and both proposed schemes are shown to impose decoding complexity, which is independent of the size of the signal set. It is observed from our simulation results that the proposed FE-DSM scheme suffers no performance loss compared with the existing DM-based DSM (DM-DSM) scheme, whereas FE-DSM-DR is observed to give a better bit-error-ratio performance at higher data rates than its DM-DSM counterpart. Specifically, at data rates of 2.25 and 2.75 bits per channel use, FE-DSM-DR is observed to achieve about 1- and 2-dB signal-to-noise ratio (SNR) gain with respect to its DM-DSM counterpart. [ABSTRACT FROM PUBLISHER]

Let F be a field, char F ≠ 2 , L / F a quartic field extension. Define by G L / F the group of elements r ∈ F ⁎ such that D ∪ ( r ) = 0 for any regular field extension K / F and any D ∈ Br 2 ( K L / K ) . We show that G L / F = F ⁎ 2 N L / F L ⁎ . As a consequence we prove that the Hasse norm theorem modulo squares holds for L / F . [ABSTRACT FROM AUTHOR]

The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs ( u , v ) satisfying u v = v u . We provide an example of a field F and an integer n such that the commuting graph of Mat n ( F ) has maximal possible diameter, equal to six. [ABSTRACT FROM AUTHOR]

Given a prime power q and a positive integer n , let F q n denote the finite field with q n elements. Also let a , b be arbitrary members of the ground field F q. We investigate the existence of a non-zero element ξ ∈ F q n such that ξ + ξ − 1 is primitive and T (ξ) = a , T (ξ − 1) = b , where T (ξ) denotes the trace of ξ in F q. This was a question intended to be addressed by Cao and Wang (2014). Their work dealt instead with another problem already in the literature. Our solution deals with all values of n ≥ 5. A related study involves the cubic extension F q 3 of F q. We show that if q ≥ 8 ⋅ 10 12 then, for any a ∈ F q , we can find a primitive element ξ ∈ F q 3 such that ξ + ξ − 1 is also a primitive element of F q 3 , and for which the trace of ξ is equal to a. This improves a result of Cohen and Gupta (2021). Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions. [ABSTRACT FROM AUTHOR]

We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an Sn degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions. [ABSTRACT FROM AUTHOR]

An elementary proof using matrix theory is given for the following criterion: if F/K and L/K are field extensions, with F and L both contained in a common extension field, then F and L are linearly disjoint over K if (and only if) some K-vector space basis of F is linearly independent over L. The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra. [ABSTRACT FROM AUTHOR]

Abstract: Let be a finite Galois extension of fields with group Γ. When Γ is nilpotent, we show that the problem of enumerating all nilpotent Hopf–Galois structures on can be reduced to the corresponding problem for the Sylow subgroups of Γ. We use this to enumerate all nilpotent (resp. abelian) Hopf–Galois structures on a cyclic extension of arbitrary finite degree. When Γ is abelian, we give conditions under which every abelian Hopf–Galois structure on has type Γ. We also give a criterion on n such that every Hopf–Galois structure on a cyclic extension of degree n has cyclic type. [Copyright &y& Elsevier]

Abstract: A direct forcing immersed boundary framework is presented for the simple and efficient simulation of strongly coupled fluid–structure interactions. The immersed boundary method developed by Yang and Balaras [J. Yang, E. Balaras, An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys. 215 (1) (2006) 12–40] is greatly simplified by eliminating several complicated geometric procedures without sacrificing the overall accuracy. The fluid–structure coupling scheme of Yang et al. [J. Yang, S. Preidikman, E. Balaras, A strongly-coupled, embedded-boundary method for fluid–structure interactions of elastically mounted rigid bodies, J. Fluids Struct. 24 (2008) 167–182] is also significantly expedited by moving the fluid solver out of the predictor–corrector iterative loop without altering the strong coupling property. Central to these improvements are the reformulation of the field extension strategy and the evaluation of fluid force and moment exerted on the immersed bodies, by taking advantage of the direct forcing idea in a fractional-step method. Several cases with prescribed motions are examined first to validate the simplified field extension approach. Then, a variety of strongly coupled fluid–structure interaction problems, including vortex-induced vibrations of a circular cylinder, transverse and rotational galloping of rectangular bodies, and fluttering and tumbling of rectangular plates, are computed. The excellent agreement between the present results and the reference data from experiments and other simulations demonstrates the accuracy, simplicity, and efficiency of the new method and its applicability in a wide range of complicated fluid–structure interaction problems. [Copyright &y& Elsevier]

Abstract: A Markov process on a local field which can be projected to a Markov process on a smaller local field is regarded as a lift of the one on the smaller field. The first part of this article is concerned with a Markov process on a local field which is obtained as the one projected from a larger field by means of the algebraic trace. Since the explicit expression of the transition probability plays important roles in a study of Markov processes on local fields, the second part is devoted to finding an explicit expression for the Markov process. [Copyright &y& Elsevier]

Abstract: In this work, we give a new and elementary proof that simultaneous similarity and simultaneous equivalence of families of matrices are invariant under extension of the ground field, a result which is non-trivial for finite fields and first appeared in an earlier paper of Klinger and Levy. [Copyright &y& Elsevier]

Abstract: A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation. [Copyright &y& Elsevier]

Let K ⊂ L be a field extension. Given K-subspaces A, B of L, we study the subspace 〈AB〉 spanned by the product set AB = {ab∣ a ∈ A, b ∈ B}. We obtain some lower bounds on dimK〈AB〉 and dimK〈Bn〉 in terms of dimK A, dimK B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson. [ABSTRACT FROM AUTHOR]

The two-dimensional local field K = F q(( u))(( t)), char K = p, and its Brauer group Br( K) are considered. It is proved that, if L = K( x) is the field extension for which we have x p − x = ut − p =: h, then the condition that ( y, f | h]K = 0 for any y ε K is equivalent to the condition f ε Im(Nm( L*)). [ABSTRACT FROM AUTHOR]

Let k0 be a field, chark0≠2, n⩾2, a,b1,…,bn∈k0* such that the elements a,b1,…,bn∈k0*/k0*2 are linearly independent. We show that there exists a field extension F/k0 and an anisotropic 4-dimensional form ϕ over F such that d±(ϕ)=a, the form ϕF(√ of b1,…,bn) is isotropic and the form (ϕF(b1,…,b1,…,√ of bn) is isotropic and the form (ϕF(b1,…,bn) is isotropic and the form (ϕF(√ of b1,…,b1,…,√ of bn))an is not defined over F. [Copyright &y& Elsevier]

Let R be a complete discrete valuation ring of mixed characteristic with perfect residue field, and let H be a finite local commutative R-Hopf algebra. We consider when there exists a finite extension of the field of fractions of R, whose valuation ring is a Galois H-object. If this occurs then H is monogenic. Conversely, if H is also cocommutative and H is monogenic, then there exists a valuation ring which is a Galois H-object. To prove this result, we represent H as the kernel of an isogeny of a special type between formal groups over R. We deduce that if A is a finite abelian R-Hopf algebra, such that both A and its dual are local, then A is the associated order of a valuation ring if and only if the dual of A is monogenic. [Copyright &y& Elsevier]

We present a construction of linear codes from polynomials. It turns out that some new codes are obtained from our construction and improve parameters of Brouwer's table [1]. [ABSTRACT FROM AUTHOR]