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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]

In this paper, we consider polynomial composites with the coefficients from K ⊂ 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. [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]

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]

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]

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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]

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]

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]

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]

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]

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]

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]

Let k ⊆ k′ be a field extension. We give relations between the kernels of higher derivations on k[ X] and k′[ X], where k[ X]:= k[ x1,…, xn] denotes the polynomial ring in n variables over the field k. More precisely, let D = { Dn} n=0∞ a higher k-derivation on k[ X] and D′ = { D′ n} n=0∞ a higher k′-derivation on k′[ X] such that D′ m( xi) = Dm( xi) for all m ⩾ 0 and i = 1, 2,…, n. Then (1) k[ X] D = k if and only if k′[ X] D′ = k′; (2) k[ X] D is a finitely generated k-algebra if and only if k′[ X] D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[ X] D of a higher derivation D of k[ X] can be generated by a set of closed polynomials. [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]

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]

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]

We consider an infinite extension K of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. K is equipped with an inductive limit topology; its conjugate K; is a completion of K with respect to a topology given by certain explicitly written semi-norms. We construct and study a Gaussian measure, a Fourier transform, a fractional differentiation operator and a cadlag Markov process on K. If we deal with Galois extensions then all these objects are Galois-invariant. [ABSTRACT FROM AUTHOR]