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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]

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]

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]

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]

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 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]

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]