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A new way to formulate the notions of minimal basis and minimal indices is developed in this paper, based on the concept of a filtration of a vector space. The goal is to provide useful new tools for working with these important concepts, as well as to gain deeper insight into their fundamental nature. This approach also readily reveals a strong minimality property of minimal indices, from which follows a characterization of the vector polynomial bases in rational vector spaces. The effectiveness of this new formulation is further illustrated by proving several fundamental properties: the invariance of the minimal indices of a matrix polynomial under field extension, the direct sum property of minimal indices, the polynomial linear combination property, and the predictable degree property.

Summary This is the third part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of ℤ n , ℚ and ℝ we have ℤ n ∩ ℤ n [X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root.

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

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.

We give a geometric interpretation of Broussous-Grabitz embedding types. We fix a central division algebra $D$ of finite index over a non-Archimedean local field $F$ and a positive integer $m$. Further we fix a hereditary order $\mathfrak{a}$ of $\operatorname{M}_m(D)$ and an unramified field extension $E|F$ in $\operatorname{M}_m(D)$ which is embeddable in $D$ and which normalizes $\mathfrak{a}$. Such a pair $(E,\mathfrak{a})$ is called an embedding. The embedding types classify the $\operatorname{GL}_m(D)$-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of $(E,\mathfrak{a})$ from the simplicial type of the image of the barycenter of $\mathfrak{a}$ under the canonical isomorphism, from the set of $E^\times$-fixed points of the reduced building of $\operatorname{GL}_m(D)$ to the reduced building of the centralizer of $E^\times$ in $\operatorname{GL}_m(D)$. Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.

Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed-Solomon and Gabidulin codes has increased lately, due to applications in distributed computations. In particular, a certain condition pertaining the intersection of zero entries at different rows, was shown to be necessary and sufficient for the existence of the sparsest possible generator matrix of Gabidulin codes over finite fields. In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero. Our proof builds upon and extends tools from the finite field case, combines them with a variant of the Schwartz-Zippel lemma over automorphisms, and provides a simple randomized construction algorithm whose probability of success can be arbitrarily close to one. In addition, potential applications for low-rank matrix recovery are discussed., Comment: ISIT'20

Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.

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

An algorithm being developed to compute geometric Galois groups of polynomials over Q( t ) is discussed. This algorithm also computes a field extension such that the factorization of a polynomial over this field extension is the absolute factorization of the polynomial.

We consider the question of certifying that a polynomial in \({\mathbb Z}[x]\) or \({\mathbb Q}[x]\) is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv. that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we assume that primes and irreducibles in \({\mathbb F}_p[x]\) are self-certifying.

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field $\mathbb{F}_p$ at once. They can also be used to represent the algebraic closure $\bar{\mathbb{F}}_p$, and to represent all finite fields in a standard manner. The most well known constructions are Conway polynomials, and the Bosma-Cannon-Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time. Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table; however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representing $\bar{\mathbb{F}}_p$. Compared to Bosma-Cannon-Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees. Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical., 9 pages, 1 figure, double column. Comments welcome!

Let G be a finite group, N and H subgroups of G with N

We classify the split simple affine algebraic groups [math] of types A and C over a field with the property that the Chow group of the quotient variety [math] is torsion-free, where [math] is a special parabolic subgroup (e.g., a Borel subgroup) and [math] is a generic [math] -torsor (over a field extension of the base field). Examples of [math] include the adjoint groups of type A. Examples of [math] include the Severi–Brauer varieties of generic central simple algebras.

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 \(k_0\) be a field of characteristic distinct from 2, \(l_0{/}k_0\) a finite field extension of degree \(2^m\), \(m\ge 2\). We prove that there exists a field extension \(K{/}k_0\) linearly disjoint with \(l_0/k_0\) such that the extension \(l_0K{/}K\) is nonexcellent. The crucial point is a construction of in some sense nonstandard elements in \(H^3(K,\mathbb {Z}{/}2\mathbb {Z})\). We apply this construction as well for investigation of the group \({F^*}^2N_{L{/}F}L^*\), where L / F is a \(\mathbb {Z}{/}4\mathbb {Z}\times \mathbb {Z}{/}2\mathbb {Z}\)-Galois extension. More precisely, let \(F\subset F_i\subset L\)\((1\le i\le 3)\) be the three intermediate quadratic extensions of F, and \(N_i(F)=N_{F_i{/}F}F_i^*\). We show that the quotient group \({N_1(F)\cap N_2(F)\cap N_3(F)\over {F^*}^2N_{L/F}L^*}\) can be arbitrarily large.

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 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 $2^k$ $\left(k\in\mathbb{Z}\right)$ vertices exhibit pretty good state transfer when there is no external dynamic control over the system. We generalize 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., arXiv admin note: text overlap with arXiv:1705.08859

For any admissible value of the parameters $n$ and $k$ there exist $[n,k]$-Maximum Rank distance ${\mathbb F}_q$-linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank distance codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes., 23 pages/post-print version

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism. These invariants give rise to easily computable criteria to check if two codes are inequivalent. We derive some concrete values and bounds for these dimension sequences for some known families of rank-metric codes, namely Gabidulin and (generalized) twisted Gabidulin codes. We then derive conditions on the length of the codes with respect to the field extension degree, such that codes from different families cannot be equivalent. Furthermore, we derive upper and lower bounds on the number of equivalence classes of Gabidulin codes and twisted Gabidulin codes, improving a result of Schmidt and Zhou for a wider range of parameters. In the end we use the aforementioned sequences to determine a characterization result for Gabidulin codes., Comment: 37 pages, 1 figure, 3 tables, extended version of arXiv:1905.11326

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.

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.

Let k be an algebraically closed field of arbitrary characteristic, let K / k {K/k} be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ℓ {\ell} , the absolute Galois group of K acts on the ℓ {\ell} -adic étale cohomology modules of X. We prove that this family of representations varying over ℓ {\ell} is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ℓ {\ell} become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

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.

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

The purpose of this note is to give a number of open problems on matching theory and their relation to the well-known results in this area. We also give a linear analogue of the acyclic matchings.

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the product of two distinct odd primes. The main advantage of this method is that it does not exclude the primes dividing the order of the Galois group, in contrast to other methods. We applied our algorithm to real cyclotomic fields of conductor $

We generalize Gabidulin codes to a large family of fields, non necessarily finite, possibly with characteristic zero. We consider a general field extension and any automorphism in the Galois group of the extension. This setting enables one to give several definitions of metrics related to the rank-metric, yet potentially different. We provide sufficient conditions on the given automorphism to ensure that the associated rank metrics are indeed all equal and proper, in coherence with the usual definition from linearized polynomials over finite fields. Under these conditions, we generalize the notion of Gabidulin codes. We also present an algorithm for decoding errors and erasures, whose complexity is given in terms of arithmetic operations. Over infinite fields the notion of code alphabet is essential, and more issues appear that in the finite field case. We first focus on codes over integer rings and study their associated decoding problem. But even if the code alphabet is small, we have to deal with the growth of intermediate values. A classical solution to this problem is to perform the computations modulo a prime ideal. For this, we need study the reduction of generalized Gabidulin codes modulo an ideal. We show that the codes obtained by reduction are the classical Gabidulin codes over finite fields. As a consequence, under some conditions, decoding generalized Gabidulin codes over integer rings can be reduced to decoding Gabidulin codes over a finite field.

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.

J. R. Sendra and C. Villarino are members of the research group ASYNACS (Ref. CCEE2011/R34)., In this extended abstract, we study the properties of ultraquadrics associated with automorphisms of the field K(\alpha )(t1, . . . , tn), defined by linear rational (with common denominator) or by polynomial (with polynomial inverse) coordinates. We conclude that ultraquadrics related to polynomial automorphisms can be characterized as varieties K−isomorphic to linear varieties, while ultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a blowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric variety. This information helps us to compute a parametrization of some ultraquadrics., Ministerio de Ciencia e Innovación

This paper presents a comprehensive study of the computation of square roots over finite extension fields. We propose two novel algorithms for computing square roots over even field extensions of the form ${\BBF_{{q^2}}}$ , with $q = {p^n}$ , $p$ an odd prime and $n \geq 1$ . Both algorithms have an associate computational cost roughly equivalent to one exponentiation in ${\BBF_{{q^2}}}$ . The first algorithm is devoted to the case when $q \equiv 1\, {\rm mod}\, 4$ , whereas the second one handles the case when $q \equiv 3\, {\rm mod}\,4$ . Numerical comparisons show that the two algorithms presented in this paper are competitive and in some cases more efficient than the square root methods previously known.

We study extensions of an algebraic flavor of Eilenberg–Mac Lane spectra in the category of ring spectra. We give a full description of the classification of these in the case of Galois extensions and partial results in the case of separable and etale ones.

Let T be an integral domain with a maximal ideal M, ϕ: T → K: = T/M the natural surjection, and R the pullback ϕ−1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x 1]]…[x n ]] to be catenarian, where each [x i ]] is fixed as either [x i ] or [[x i ]]. We also give a complete answer to the question of determining the field extensions k ⊂ K such that the contraction map Spec(K[x 1]]…[x n ]]) → Spec(k[x 1]]…[x n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x 1]]…[x n ]] is catenarian.

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prufer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prufer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prufer domain. Moreover, we show that any two extensions of a semilocal Prufer domain are incomparable (by inclusion) in an algebraic extension of fields.

Let (V, h) be a Hermitian space over a division algebra D which is of index at most two over a non-Archimedean local field k of residue characteristic not 2. Let G be the unitary group defined by h and let \({\sigma}\) be the adjoint involution. Suppose we are given two \({\sigma}\)-invariant but not \({\sigma}\)-fixed field extensions E1 and E2 of k in EndD(V) which are isomorphic under conjugation by an element g of G and suppose that there is a point x in the Bruhat–Tits building of G which is fixed by \({E_1^{\times}}\) and \({E_2^{\times}}\) in the reduced building of AutD(V). Then E1 is conjugate to E2 under an element of the stabilizer of x in G if E1 and E2 are conjugate under an element of the stabilizer of x in AutD(V) and a weak extra condition holds. In addition, in many cases the conjugation by g from E1 to E2 can be realized as conjugation by an element of the stabilizer of x in G. Further we give a concrete description of the canonical isomorphism from the set of \({E_1^\times}\) fixed points of the building of G onto the building of the centralizer of E1 in G.

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.

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.

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 $X$ be a quasi-compact, separated scheme over a field k and we can consider the categorical resolution of singularities of $X$. In this paper let $k^{\prime}/k$ be a field extension and we study the scalar extension of a categorical resolution of singularities of $X$ and we show how it gives a categorical resolution of the base change scheme $X_{k^{\prime}}$. Our construction involves the scalar extension of derived categories of DG-modules over a DG algebra. As an application we use the technique of scalar extension developed in this paper to prove the non-existence of full exceptional collections of categorical resolutions for a projective curve of genus $\geq 1$ over a non-algebraically closed field., 12 pages; Proposition 2.4 and Lemma 4.5 added, the proofs of Proposition 3.3 and Proposition 4.6 (previously Proposition 4.5) revised; to appear in Journal of Pure and Applied Algebra

The -transformation of the unit interval is de ned by T (x) := x (mod 1). Its eventually periodic points are a subset of [0; 1] intersected with the eld extension Q( ). If > 1 is an algebraic integer of degree d > 1, then Q( ) is a Q-vector space isomorphic to Q d , therefore the intersection of [0; 1] with Q( ) is isomorphic to a domain in Q d . The transformation from this domain which is conjugate to the -transformation is called the companion map, given its connection to the companion matrix of 's minimal polynomial. The companion map and the proposed notation provide a natural setting to reformulate a classic result concerning the set of periodic points of the -transformation for Pisot numbers. It also allows to visualize orbits in a d-dimensional space. Finally, we refer connections with arithmetic codings and symbolic representations of hyperbolic toral automorphisms.

An elliptic curve E defined over a finite field K, E(K) is the set of solutions to the general Weierstrass polynomial E: y 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6 where the coefficients a1, a2, a3, a4, a6 є K. There exist a well defined addition of points on each curve such that the points form an abelian group under the addition operation. This group is either cyclic or isomorphic to the product of two cyclic groups. These set of solutions that form the group lie in the closure of the field K over which the curve is defined. If we allow the set to lie only in a particular extension of K, the addition operation is well defined there too. Therefore we can associate a group to every extension K' of the field K denoted by E(K'). Will the structure of the group defined over the base field K, be affected if the same group is made to lie in the extension K' of K?

A polynomial source of randomness over F n is a random variable X = f(Z) where f is a polynomial map and Z is a random variable distributed uniformly over F r for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f , and the base-q logarithm of the size of the range of f over inputs in F r , denoted by k. For simplicity we call X a (q; D; k)-source.

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.

This paper discusses the question of optimizing AES hardware designs, by using the composite field representation GF(24)2 of the field GF(28), that underlies the definition of AES. Here, GF(24)2 is the field extension of the ground field GF(28) with an extension polynomial of the form x2+aamp;#x03B1;x+aamp;#x03B2;, where aamp;#x03B1; and aamp;#x03B2; are elements of field GF(24). Previous designs with such representations used aamp;#x03B1; = 1, which seemingly leads to some obvious savings. By contrast, we seek the optimal designs amongall the possibilities. Our designs are based on mapping the input, output, round keys, and the AES operations to and from any one of the 2880 possible representations of (28) as (24)2. For each representation, we also explore three options for the affine/invaffine constants, resulting in a total of 8640 possible designs. We identify the smallest area representations for AES encryption-only, decryption-only, and for unified encryption-decryption. Surprisingly, the optimal representations in each case are different from each other. In addition, we identify six distinct representations that are optimal, based on operating-mode and AES pipeline depth. Among other results, we show here a set of high-bandwidth 16-byte AES datapaths with the extension polynomials of the form x2+aamp;#x03B1;x+aamp;#x03B2; where aamp;#x03B1; aamp;#x2260; 1, showing that the a-priori obvious choice of using aamp;#x03B1; = 1, does not necessarily lead to the best result. We provide the full details of all the designs possibilities, together with their respective area, based on 22nm CMOS implementation.

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.

Essential dimension is an invariant of algebraic groups G over a field F that measures the complexity of G-torsors over field extensions of F. We use theorems of N. Karpenko about the incompressibility of Severi-Brauer varieties and quadratic Weil transfers of Severi-Brauer varieties to compute the essential dimension of some closed subgroups of RK/F (GL1(A)), where A is a central division K-algebra of prime power degree and K/F is a separable field extension of degree ≤ 2. In particular, we determine the essential dimension of the group Sim(A, σ) of similitudes of (A, σ), where σ is an F-involution on A, and the essential dimension of the normalizer \(N_{GL_1 (A)} \left( {GL_1 \left( B \right)} \right)\), where B is a separable subalgebra of 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 .