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Let A = F q [ T ] be the polynomial ring over F q , and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r ≥ 2 over F. For all but finitely many primes p ◁ A , one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ ⊗ F p of rank r over F p = A / p . The endomorphism ring E p = End F p ( ϕ ⊗ F p ) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K, and let π p ∈ E p be the Frobenius endomorphism of ϕ ⊗ F p . Then we have the inclusion of orders A [ π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg ( ϕ ) = A , then for arbitrary non-zero ideals n , m of A there are infinitely many p such that n divides the index χ ( E p / A [ π p ] ) and m divides the index χ ( O p / E p ) . We show that the index χ ( E p / A [ π p ] ) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r = 2 case we describe an algorithm for computing the orders A [ π p ] ⊂ E p ⊂ O p , and give some computational data.

A fundamental result of Springer says that a quadratic form over a field of characteristic ≠ 2 is isotropic if it is so after an odd degree field extension. In this paper we generalize Springer’s theorem as follows. Let R be an arbitrary semilocal ring, let S be a finite R -algebra of constant odd degree, which is etale or generated by one element, and let q be a nonsingular R -quadratic form whose base ring extension q S is isotropic. We show that then already q is isotropic.

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of Π Σ ⁎ -fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of Π Σ ⁎ -field extensions over a difference field that enjoys certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions of a homogeneous linear difference equation that are defined over the arising sums and products.

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.

It is an open problem whether the Iwasawa λ-invariants of the Z p -extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Z p -extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Z p -extensions.

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding skew left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K × . We consider Kummer extensions of G of the form K ( ζ 2 m , G 2 n ) / K ( ζ 2 m ) , where n ⩽ m . In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K ( ζ 4 ) . We prove how properties of G over K explicitly determine the divisibility parameters over K ( ζ 4 ) . This result yields a clear computational advantage, since no field extension is required.

Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kahler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E , in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C . We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤4 which are defined over R and over Q .

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity., Comment: Slides available at: http://www.math.fsu.edu/~hoeij/2017/Presentation.pdf

Let A be a biquaternion division algebra over a field F ( char F ≠ 2 ). We prove that an element e ∈ F ⁎ such that A F ( e 4 ) = 0 exists if and only if there is a decomposition A ≃ ( a , b ) ⊗ ( c , d ) with 《 − a , b , c , d 》 = 0 . This statement strengthens the result of Rost, Serre and Tignol ( [7] ), where the additional condition − 1 ∈ F ⁎ holds. Another result is a construction of a field F with − 1 ∈ F ⁎ , a quartic cyclic field extension L / F , and quaternion algebras ( a , b ) and ( c , d ) over F such that ( a , b ) L = ( c , d ) L = 0 , but the quadratic form 〈 a , b , a b 〉 ⊥ 《 c , d 》 is anisotropic. As a consequence, for any algebraically closed field k 0 , we give an example of an anisotropic 7-dimensional quadratic form Ψ over k 0 ( z , x , y ) with ind ( C 0 ( Ψ ) ) = 4 such that for any discrete k 0 ( z ) -valuation v on k 0 ( z , x , y ) the form Ψ becomes isotropic over the completion k 0 ( z , x , y ) v . In other words, the form Ψ provides a counterexample to the strong Hasse principle for 7-dimensional quadratic forms with respect to the rational extension K ( x , y ) / K , where K = k 0 ( z ) .

Let O be a differential graded (possibly colored) operad defined over rationals. Let us assume that there exists a zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology , where K is any field extension of Q . We show that for a large class of such dg operads, a formality quasi-isomorphism for O exists and can be constructed recursively. Every step of our recursive procedure involves a solution of a finite dimensional linear system and it requires no explicit knowledge about the zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology.

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.

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial degree $s$, independent of the underlying field extension degree~$m$. We show that our multiplication algorithm is faster than all known ones when $s \leq m$. Using a result by Caruso and Le Borgne (2017), this immediately implies a sub-quadratic division algorithm for linearized polynomials for arbitrary polynomial degree $s$. Also, we propose algorithms with sub-quadratic complexity for the $q$-transform, multi-point evaluation, computing minimal subspace polynomials, and interpolation, whose implementations were at least quadratic before. Using the new fast algorithm for the $q$-transform, we show how matrix multiplication over a finite field can be implemented by multiplying linearized polynomials of degrees at most $s=m$ if an elliptic normal basis of extension degree $m$ exists, providing a lower bound on the cost of the latter problem. Finally, it is shown how the new fast operations on linearized polynomials lead to the first error and erasure decoding algorithm for Gabidulin codes with sub-quadratic complexity., Comment: 25 pages, submitted to Journal of Symbolic Computation

Recently, Bollen, Draisma, and Pendavingh have introduced the Lindstr\"om valuation on the algebraic matroid of a field extension of characteristic p. Their construction passes through what they call a matroid flock and builds on some of the associated theory of matroid flocks which they develop. In this paper, we give a direct construction of the Lindstr\"om valuated matroid using the theory of inseparable field extensions. In particular, we give a description of the valuation, the valuated circuits, and the valuated cocircuits., Comment: 12 pages, v2: added Section 3 on valuated cocircuits and minors, v3: minor changes and corrections, final submitted version

Motivated by the analysis of neural net models between integer and rational weights, we introduce a so-called cut language over a real digit alphabet, which contains finite β-expansions (i.e. base-β representations) of the numbers less than a given threshold. We say that an infinite β-expansion is eventually quasi-periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits, contains an infinite constant subsequence. We prove that a cut language is regular iff its threshold is a quasi-periodic number whose all β-expansions are eventually quasi-periodic, by showing that altogether they have a finite number of tail values. For algebraic bases β, we prove that there is an eventually quasi-periodic β-expansion with an infinite number of tail values iff there is a conjugate of β on the unit circle. For transcendental β combined with algebraic digits, a β-expansion is eventually quasi-periodic iff it has a finite number of tail values. For a Pisot base β and digits from the smallest field extension Q ( β ) over rational numbers including β, we show that any number from Q ( β ) is quasi-periodic. In addition, we achieve a dichotomy that a cut language is either regular or non-context-free and we show that any cut language with rational parameters is context-sensitive.

We investigate Hopf–Galois structures on a cyclic field extension L / K of squarefree degree n. By a result of Greither and Pareigis, each such Hopf–Galois structure corresponds to a group of order n, whose isomorphism class we call the type of the Hopf–Galois structure. We show that every group of order n can occur, and we determine the number of Hopf–Galois structures of each type. We then express the total number of Hopf–Galois structures on L / K as a sum over factorisations of n into three parts. As examples, we give closed expressions for the number of Hopf–Galois structures on a cyclic extension whose degree is a product of three distinct primes. (There are several cases, depending on congruence conditions between the primes.) We also consider one case where the degree is a product of four primes.

We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rk_F for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rk_F depends on rk continuously. When an amenable group has a twisted action on a unital C*-algebra preserving a tracial state, we also show that two natural Sylvester matrix rank functions on the algebraic twisted crossed product constructed out of the tracial state coincide., Comment: Proposition 9.6 is added. 45 pages. To appear in J. Funct. Anal

An n × n matrix pattern is said to be spectrally arbitrary over a field F provided for every monic polynomial p ( t ) of degree n, with coefficients from F , there exists a matrix with entries from F , in the given pattern, that has characteristic polynomial p ( t ) . Let E ⊆ F ⊆ K be an extension of fields. It is natural to ask whether a pattern that is spectrally arbitrary over F must also be spectrally arbitrary over E or K . In this article it is shown that if F is dense in K and K is a complete metric space, then any spectrally arbitrary or relaxed spectrally arbitrary pattern over F is relaxed spectrally arbitrary over K . It is also established that if E is an algebraically closed subfield of a field F , then any spectrally arbitrary pattern over F is spectrally arbitrary over E . The 2n Conjecture and the Superpattern Conjecture are explored over fields other than the real numbers. In particular, examples are provided to show that the Superpattern Conjecture is false over the field with 3 elements.

Let G be a split simple affine algebraic group of type A or C over a field k, and let E be a standard generic G-torsor over a field extension of k. We compute the Chow ring of the variety of Borel subgroups of G (also called the variety of complete flags of G), twisted by E. In most cases, the answer contains a large finite torsion subgroup. The torsion-free cases have been treated in the predecessor Chow ring of some generically twisted flag varieties by the author.

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 .

Let T/A be an integral extension of noetherian integrally closed integral domains whose quotient field extension is a finite cyclic Galois extension. Let S/R be a localization of this extension which is unramified. Using a generalized cyclic crossed product construction it is shown that certain reflexive fractional ideals of T with trivial norm give rise to Azumaya R-algebras that are split by S. Sufficient conditions on T/A are derived under which this construction can be reversed and the relative Brauer group of S/R is shown to fit into the exact sequence of Galois cohomology associated to the ramified covering T/A. Many examples of affine algebraic varieties are exhibited for which all of the computations are carried out.

We extend the concept of the difference dimension polynomial of a difference field extension to difference local algebras. The main theorem of the paper establishes the existence and form of the dimension polynomial associated with the localization of a finitely generated well-mixed difference algebra at a prime reflexive difference ideal.

Let F be a field of characteristic p > 0 . Let Ω n ( F ) be the F-vector space of n-differentials of F over F p . Let K = F ( g ) be the function field of an irreducible polynomial g in m ⩾ 1 variables over F. We derive an explicit description of the kernel of the restriction map Ω n ( F ) → Ω n ( K ) . As an application in the case p = 2 , we determine the kernel of the restriction map when passing from the Witt ring (resp. graded Witt ring) of symmetric bilinear forms over F to that over such a function field extension K.

Let F be a field of characteristic 2 and let E / F be a field extension of degree 4. We determine the kernel W q ( E / F ) of the restriction map W q F → W q E between the Witt groups of nondegenerate quadratic forms over F and over E, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduce the corresponding result for the Witt kernel W ( E / F ) of the restriction map W F → W E between the Witt rings of nondegenerate symmetric bilinear forms over F and over E from earlier results by the first author. As an application, we describe the 2-torsion part of the Brauer kernel for such extensions.

Let k be field of characteristic ≠ 2, 3. Assume that k contains more than 11 elements and let K be an algebraic field extension of k. The present paper describes subgroups of the general linear group GL 7 ( K ) that contain the Chevalley group G 2 ( k ) .

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=ap^{\nu}$. The indices of inseparability $i_0,i_1,...,i_{\nu}$ of $L/K$ were defined by Fried in the case char$(K)=p$ and by Heiermann in the case char$(K)=0$; they give a refinement of the usual ramification data for $L/K$. The indices of inseparability can be used to construct "generalized Hasse-Herbrand functions" $\phi_{L/K}^j$ for $0\le j\le\nu$. In this paper we give an interpretation of the values $\phi_{L/K}^j(c)$ for natural numbers $c$. We use this interpretation to study the behavior of generalized Hasse-Herbrand functions in towers of field extensions., Comment: 16 pages

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integer coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a “categorification” of the classical corestriction homomorphism between Brauer groups.

In this paper, we present an overview of some of our research on the Levi-Civita fields R and C . R (resp. C ) is the smallest non-Archimedean field extension of the real (resp. complex) numbers that is Cauchy-complete and real closed (resp. algebraically closed); in fact, R is small enough to allow for the calculus on the field to be implemented on a computer and used in applications such as the fast and accurate computation of the derivatives of real functions as "differential quotients" up to very high orders. We summarize the convergence and analytical properties of power series, showing that they have the same smoothness behavior as real and complex power series; we present a Lebesgue-like measure and integration theory on the Levi-Civita field R ; we discuss solutions to one-dimensional and multi-dimensional optimization problems based on continuity and differentiability concepts that are stronger than the topological ones; and we give a brief summary of the results of our ongoing work on developing a non-Archimedean operator theory on a Banach space over C .

Let S be a semianalytic algebra over a field k of characteristic p > 0 and S p the local ring with respect to p ∈ Spec ( S ) . In this article, for a semianalytic algebra we introduce the concept of semianalytic p n -basis which generalizes the p n -basis defined in [1] , and the concept of a p n -admissible field for a semianalytic field extension, we give regularity criteria for the ring S p without reducibility assumption. The results are extension of our previous results for affine algebra to the case of semianalytic algebra (cf. [1] ), and these are generalization of results of Orbanz [6] in the semianalytic case of prime characteristic.

Let L/K be a separable field extension of degree 6. An 1867 theorem of P. Joubert asserts that if char(K) is different from 2 then L is generated over K by an element whose minimal polynomial is of the form t^6 + a t^4 + b t^2 + ct + d for some a, b, c, d in K. We show that this theorem fails in characteristic 2., 5 pages, new material added

Let O be the ring of integers of a non-Archimedean local field and π a fixed uniformizer of O. We prove that the exterior powers of a π -divisible O-module scheme of dimension at most 1 over a field exist and commute with field extensions. We calculate the height and the dimension of the exterior powers in terms of the height of the given π -divisible O-module scheme.

All counterexamples of Pinchuk type to the strong real Jacobian conjecture are shown to have rational function field extensions of degree 6 with no nontrivial automorphisms.

In a previous piece of work it has been shown that the exponential stability of a linear time-varying (LTV) system can be evaluated using new definitions of the poles of such a system. The latter are given by a fundamental set of roots of the skew polynomial P ( ∂ ) which defines the autonomous part of the system. Such a set may not exist over the initial field K of definition of the coefficients of the system, but can exist over a suitable field extension K ⊃ K . It is shown here that conditions for stability can also be obtained using linear factors of the polynomial P ( ∂ ) over another field extension K which may be smaller: K ⊃ K ⊃ K . The roots of these factors are called the quasi-poles of the system. The necessary condition for system stability, expressed in function of these quasi-poles, is more restrictive than the one involving a fundamental set of roots.

Given a field extension K/k of degree n we are interested in finding the subfields of K containing k. There can be more than polynomially many subfields. We introduce the notion of generating subfields, a set of up to n subfields whose intersections give the rest. We provide an efficient algorithm which uses linear algebra in k or lattice reduction along with factorization in any extension of K. Implementations show that previously difficult cases can now be handled.

Let L / K 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 L / K 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 L / K 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.

Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi–Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E / F , the motivic decompositions of several Severi–Brauer varieties of D remain the same when extending the scalars to E.

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory., 11 pages

We study the lattice L of subgroups of a Chevalley group G(Φ,A) over a ring A, containing its elementary subgroup E(Φ,S) over a subring S⊆A. The standard description asserts that for any H∈L there exists a unique subring R⊇S of A such that H contains E(Φ,R) as a normal subgroup.The standard description was obtained by Ya. Nuzhin for algebraic field extensions. On the other hand, it was expected that the standard description does not hold for transcendental extensions. This was recently proved by the author for simply laced root systems.Now, suppose that Φ is doubly laced, i.e. Φ=Bl, Cl or F4, and that 2 is invertible in S. In the article we prove that in these settings the standard description of L holds for an arbitrary pair of rings S⊆A.

1. Let A 1 , … , A n be central simple disjoint algebras over a field F . Let also l i | exp ( A i ) , m i | ind ( A i ) , l i | m i , and for each i = 1 , … , n , let l i and m i have the same sets of prime divisors. Then there exists a field extension E / F such that exp ( A i E ) = l i and ind ( A i E ) = m i , i = 1 , … , n . 2. Let A be a central simple algebra over a field K with an involution τ of the second kind. We prove that there exists a regular field extension E / K preserving indices of central simple K -algebras such that A ⊗ K E is cyclic and has an involution of the second kind extending τ .

For a nondegenerate quadratic form phi on a vector space V of dimension 2n + 1, let X_d be the variety of d-dimensional totally isotropic subspaces of V. We give a sufficient condition for X_2 to be 2-incompressible, generalizing in a natural way the known sufficient conditions for X_1 and X_n. Key ingredients in the proof include the Chernousov-Merkurjev method of motivic decomposition as well as Pragacz and Ratajski's characterization of the Chow ring of (X_2)_E, where E is a field extension splitting phi., 12 pages

The linear- or affine-invariance is the property of a function family that is closed under linear- or affine-transformations on the domain, and closed under linear combinations of functions, respectively. Both the linear- and affine-invariant families of functions are generalizations of many symmetric families, for instance, the low degree polynomials. Kaufman and Sudan (2007) [21] introduced the notions of ''constraint'' and ''characterization'' to characterize the locally testable affine- and linear-invariant families of functions over finite fields of constant size. In this article, it is shown that, for any finite field F of size q and characteristic p, and its arbitrary extension field K of size Q, if an affine-invariant family @?@?{K^n->F} has a k-local constraint, then it is k^'-locally testable for k^'=k^2^Q^pQ^2^Q^p^+^4; and that if a linear-invariant family @?@?{K^n->F} has a k-local characterization, then it is k^'-locally testable for k^'=2k^2^Q^pQ^4^(^Q^p^+^1^). Consequently, for any prime field F of size q, any positive integer k, we have that for any affine-invariant family @? over field F, the four notions of ''the constraint'', ''the characterization'', ''the formal characterization'' and ''the local testability'' are equivalent modulo a poly(k,q) of the corresponding localities; and that for any linear-invariant family, the notions of ''the characterization'', ''the formal characterization'' and ''the local testability'' are equivalent modulo a poly(k,q) of the corresponding localities. The results significantly improve, and are in contrast to the characterizations in [21], which have locality exponential in Q, even if the field K is prime. In the research above, a missing result is a characterization of linear-invariant function families by the more natural notion of constraint. For this, we show that a single strong local constraint is sufficient to characterize the local testability of a linear-invariant Boolean function family, and that for any finite field F of size q greater than 2, there exists a linear-invariant function family @? over F such that it has a strong 2-local constraint, but is not q^d^q^-^1^-^1-locally testable. The proof for this result provides an appealing approach toward more negative results in the theme of characterization of locally testable algebraic properties, which is rare, and of course, significant.

In this paper, we formulate and prove linear analogues of results concerning matchings in groups. A matching in a group G is a bijection f between two finite subsets A,B of G with the property, motivated by old questions on symmetric tensors, that the product af(a)does not belong to A for all a \in A. Necessary and sufficient conditions on G, ensuring the existence of matchings under appropriate hypotheses, are known. Here we consider a similar question in a linear setting. Given a skew field extension K \subset L, where K commutative and central in L, we introduce analogous notions of matchings between finite-dimensional K-subspaces A,B of L, and obtain existence criteria similar to those in the group setting. Our tools mix additive number theory, combinatorics and algebra., Comment: The present version corrects a slight gap in the statement of Theorem 2.6 of the published version of this paper [Journal of Algebra 324 (2010) 3420-3430]

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 a paper of Klinger and Levy., Comment: 10 pages (minor corrections)

We define a class of algebras over finite fields, called polynomially cyclic algebras, which extend the class of abelian field extensions. We study the structure of these algebras; furthermore, we define and investigate properties of Lagrange resolvents and Gauss and Jacobi sums.Natural examples of polynomially cyclic algebras are for instance algebras of the form Fp[X]/(Fq(X)) where p,q are distinct odd primes and Fq is the q-th cyclotomic polynomial. Further examples occur similarly on replacing the cyclotomic polynomials with factors of division polynomials of elliptic curves. Finally, Gauss and Jacobi sums over polynomially cyclic algebras are applied for improving current algorithms for counting the number of points of elliptic curves over finite fields.

In the first part of this paper we compute the Witt ring kernel for an arbitrary field extension of degree 4 and characteristic different from 2 in terms of the coefficients of a polynomial determining the extension. In the case where the lower field is not formally real we prove that the intersection of any power n of its fundamental ideal and the Witt ring kernel is generated by n -fold Pfister forms. In the second part as an application of the main result we give a criterion for the tensor product of quaternion and biquaternion algebras to have zero divisors. Also we solve the similar problem for three quaternion algebras. In the last part we obtain certain exact Witt group sequences concerning dihedral Galois field extensions. These results heavily depend on some similar cohomological results of Positselski, as well as on the Milnor conjecture, and the Bloch–Kato conjecture for exponent 2, which was proven by Voevodsky.

Let F be a field of characteristic not 2, a 1 , … , a n ∈ F ∗ , M = F ( a 1 , … , a n ) . Let further G be the subgroup of F ∗ / F ∗ 2 generated by a 1 , … , a n . For any I ⊂ { 1 , … , n } put a I = ∏ i ∈ I a i ( a ∅ = 1 ) . Consider the complex ∏ I ≠ ∅ K I ∗ / K I ∗ 2 → φ G ⊗ F ∗ / F ∗ 2 → ψ Br 2 F → res Br 2 M → ∏ N M / E ∏ F ⊂ E ⊂ M , [ M : E ] = 2 Br 2 E , where K I = F ( a I ) are all the quadratic extensions of F , containing in M , φ ( { w I } ) = ∑ I a I ⊗ N K I / F ( w I ) , ψ ( a I ⊗ z ) = ( a I , z ) . The homology group of this complex at i + 1 -th term from the left is denoted by h i ( M / F ) . Independently of char F we give examples of these complexes with nontrivial h 3 ( M / F ) and prescribed n ⩾ 3 and a 1 , … , a n − 1 . Also similar examples concerning the group h 1 ( M / F ) are constructed.

We extend the classical rigidity results for K -theory to the equivariant setting of linear algebraic group actions. These results concern rigidity for rational points, field extensions, and Hensel local rings. To cite this article: S. Yagunov, P.A. Ostvaer, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

We give an overview on twisting commutative algebraic groups and applications to discrete log-based cryptography. We explain how discrete log-based cryptography over extension fields can be reduced to cryptography in primitive subgroups. Primitive subgroups in turn are part of a general theory of tensor products of commutative algebraic groups and Galois modules (or twists of commutative algebraic groups), and this underlying mathematical theory can be used to shed light on discrete log-based cryptosystems. We give a number of concrete examples, to illustrate the definitions and results in an explicit way.

We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite provided that the ring is either right noetherian or countably generated over a large field. We give examples to show that, over countable fields, a just infinite algebra which is either affine or non-noetherian need not remain just infinite under extension of scalars. We also give a concrete classification of PI stably just infinite rings and give two characterizations of non-PI stably just infinite rings in terms of Martindale's extended center., Comment: fixed typos

Let G be a finite group and R G be its group algebra defined over R . If we define in G a 2-cochain F, then we can consider the algebra R F G which is obtained from R G deforming the product, x . F y = F ( x , y ) x y , ∀ x , y ∈ G . Examples of R F ( Z 2 ) n algebras are Clifford algebras and Cayley algebras like octonions. In this paper we consider generalizations of lattices with complex multiplication in the context of these twisted group algebras. We explain how these induce the natural algebraic structure to endow any arbitrary finite-dimensional lattice whose real components stem from any finite algebraic field extension over Q with a multiplicative closed structure. Furthermore, we develop some fully explicit characterizations in terms of generalized trace and norm functions.