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Let A p ⊂ C denote the set of all algebraic numbers such that α ∈ A p if and only if α is a zero of a (not necessarily irreducible) polynomial with positive rational coefficients. We give several results concerning the numbers in A p . In particular, the intersection of A p and the unit circle |z| = 1 is investigated in detail. So we determine all numbers of degree less than 6 on the unit circle which lie in the set A p . Further we show that when α is a root of an irreducible rational polynomial p(X) of degree ≠ 4 whose Galois group contains the full alternating group, α lies in A p if and only if no real root of p(X) is positive.

We previously described the Galois module structures of pth-power class groups K×/K×p, where K/F is a cyclic extension of degree p over a field F containing a primitive pth root of unity. That description relied upon arithmetic invariants associated with K/F. Here we construct field extensions K/F with prescribed arithmetic invariants, thus completing the classification of Galois modules K×/K×p.

Let k be a finite field, a global field, or a local non-archimedean field, and let H 1 and H 2 be split, connected, semisimple algebraic groups over k. We prove that if H 1 and H 2 share the same set of maximal k-tori, up to k-isomorphism, then the Weyl groups W(H 1 ) and W(H 2 ) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B n and C n . (Due to a recent result of Philippe Gille, this result also holds for fields which admit arbitrary cyclic extensions.).

In this Note we give a counter example to a conjecture of Malle which predicts the asymptotic behavior of the counting functions for field extensions with given Galois group and bounded discriminant. To cite this article: J. Kluners, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

In this paper, we study the bounded sum-of-digits discrete logarithm problem in finite fields. Our results are concerned primarily with fields Fqn, where n|q - 1. The fields are called Kummer extensions of Fq. It is known that we can efficiently construct an element g with order exponential in n. Let $S_q(\bullet)$ be the function from integers to the sum of digits in their q-ary expansions. We first present an algorithm that, given ge (0 $\leq$ e < qn), finds e in random polynomial time, provided that Sq (e) < n. We then show that the problem is solvable in random polynomial time for most of the exponent e with Sq (e) < 1.32 n by exploring an interesting connection between the discrete logarithm problem and the problem of list decoding of Reed--Solomon codes and applying the Guruswami--Sudan algorithm. As far as we are aware, our algorithm is the first one which can solve discrete logarithms of $2^{\log^{1-\epsilon}{q^n}}$ many instances in polynomial time for infinite many constant characteristic fields Fqn. Furthermore, since every finite field has an extension of reasonable degree, which is a Kummer extension, our result reveals an unexpected property of the discrete logarithm problem, namely, the bounded sum-of-digits discrete logarithm problem in any given finite field becomes polynomial-time solvable in certain low degree extensions. As a side result, we obtain a sharper lower bound on the number of congruent polynomials generated by linear factors than the one based on the Stothers--Mason ABC-theorem. We also prove that, in the field Fqq-1, the bounded sum-of-digits discrete logarithm with respect to g can be computed in random time O(f(w)log4 (qq-1)), where f is a subexponential function and w is the bound on the q-ary sum-of-digits of the exponent; hence the problem is fixed parameter tractable. These results are shown to be generalized to Artin--Schreier extension Fpp, where p is a prime.

Let K be a field, L be a field extension of K, and G be a connected semisimple algebraic group defined over K. In [6], we called a K-torus T to be K-irreducible if it does not contain any proper K-subtori. We established the existence of such tori in any connected absolutely (almost) simple algebraic group over a global field and gave applications of this result, in particular, to the congruence subgroup problem. Subsequently, in [8] we considered maximal tori in arbitrary semisimple groups, where the notion of irreducible tori had to be replaced with a more general notion of quasi-irreducible tori. As in [8], we will say that a maximal L-torus T of G is L-quasi-irreducible (or, quasi-irreducible over L) if it does not contain any L-subtori other than (almost direct) products of the subtori T (i) := T ∩G(i), where G, . . . , G are the connected L-simple normal subgroups of G. In [8], we proved the existence of K-quasi-irreducible tori in G if K is a finitely generated field of characteristic zero. (A noteworthy feature of the argument in [8] is that it is based on the technique of embedding relevant finitely generated fields into nonarchimedean local fields, developed and used earlier in [7] for a different purpose.) Assume now that K is a finitely generated subfield of the field R of real numbers. In [8] we proved that any Zariski-dense subsemigroup of G(K) contains a regular R-regular element x such that the centralizer T of x in G is a K-quasiirreducible torus and the cyclic subgroup of T (K) generated by x is Zariski-dense in T . Since such elements will also play an important role in the current paper, we recall for the reader’s convenience that a semisimple element x ∈ G is called regular if the identity component ZG(x)◦ of its centralizer is a (maximal) torus; note that x ∈ ZG(x)◦ ([4], 11.12). Regular elements form a Zariski-open subset of G (cf. [4], §12). A regular element x of G(L) will be called L-quasi-irreducible if the maximal L-torus T := ZG(x)◦ is L-quasi-irreducible, and x will be called L-quasi-irreducible anisotropic if, in addition, T is anisotropic over L. We will say that an element x ∈ G(K) is without components of finite order, if in some (equivalently, any) decomposition x = x1 · · ·xt, with xi ∈ Gi, where G1, . . . ,Gt are the connected absolutely simple normal subgroups of G, all the xi’s have infinite order. It is easy to see that (for x ∈ G(K)) this notion is equivalent to the one introduced in [8]. An element x ∈ G(R) is called R-regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad x is minimum possible. Such an element

Let F be a field, char F ≠ 2, and ϕ and ψ be anisotropic quadratic forms over F. Let L be a generic field extension of F such that i(ψ L ) ≥ 2. Under what conditions is the form ϕ L isotropic? We give an answer to this question in the cases where dim ϕ = 5, dim ψ = 6 and dim ϕ = 6, dim ψ = 7. Bibliography: 18 titles.

We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms, the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n=d/sup k/ point multiplier with area that is subquadratic and delay that is logarithmic in the bit-length n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method, we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm.

Let $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

The paper has two parts. First we prove that the specialization maps on R-equivalence and on the Chow group of zero cycles are isomorphisms for families over a local, Henselian, Dedekind ring when the special fiber is smooth and separably rationally connected. This is pretty much the best one can hope for. The second part contains examples of unirational quartic hypersurfaces whith infinitely many R-equivalence classes. There are also examples of quartics with points in a large odd degree field extension but no points in any smaller odd degree extension.

We show that if a field k contains sufficiently many elements (for instance, if k is infinite), and K is an algebraically closed field containing k, then every linear algebraic k-group over K is k-isomorphic to Aut(A ⊗k K), where A is a finite dimensional simple algebra over k. In this paper, ‘algebra’ over a field means ‘nonassociative algebra’, i.e., av ector space A over this field with multiplication given by a linear map A ⊗ A → A, a1 ⊗ a2 � a1a2, subject to no a priori conditions; cf. [Sc]. Fix a field k and an algebraically closed field extension K/k. Our point of view of algebraic groups is that of [Bor], [H], [Sp]; the underlying varieties of linear algebraic groups will be the affine algebraic varieties over K .A susual, algebraic group (resp., subgroup, homomorphism) defined over k is abbreviated to k-group (resp., k-subgroup, k-homomorphism). If E/F is a field extension and V is a vector space over F ,w edenote by VE the vector space V ⊗F E over E. Let A be a finite dimensional algebra over k and let V be its underlying vector space. The k-structure V on VK defines the k-structure on the linear algebraic group GL(VK). As Aut(AK), the full automorphism group of AK, is a closed subgroup of GL(VK), it has the structure of a linear algebraic group. If Aut(AK )i sdefined over k (that is always the case if k is perfect; cf. [Sp, 12.1.2]), then for each field extension F/k the full automorphism group Aut(AF )o fF -algebra AF is the group Aut(AK)(F )o fF -rational points of the algebraic group Aut(AK).

Let D be a principal ideal domain with quotient field F and suppose every residue field of D is finite. Let K be a finite separable field extension of F of degree at least 4 and let [Formula: see text] denote the integral closure of D in K. Let [Formula: see text] where f ∈ D is a nonzero nonunit. In this paper we show, assuming a mild condition on f, that cancellation of finitely generated modules fails for R, that is, there exist finitely generated R-modules L, M, and N such that L ⊕ M ≅ L ⊕ N and yet M ≇ N. In case the unit group of D is finite, we show that cancellation fails for almost all rings of the form [Formula: see text], where p ∈ D is prime.

We modify the abc-conjecture for number fields K in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent μ > 1 of the absolute value D K of the discriminant cannot be replaced by μ = 1, and even that an arbitrarily large power of log D K must be present.

LetG=GL(m, D) whereD is a central division algebra over a commutative nonarchimedean local fieldF. LetE/F be a field extension contained inM(m, D). We denote byI (resp.IE) the nonextended affine building ofG (resp. of the centralizer ofEx inG). In this paper we prove that there exists a uniqueGE-equivariant affine mapjE∶IE→I. It is injective and its image coincides with the set ofEx-fixed points inI. Moreover, we prove thatjE is compatible with the Moy-Prasad filtrations.

Let [Formula: see text] be a field, [Formula: see text], [Formula: see text][Formula: see text], [Formula: see text] a quartic field extension. We investigate the divided power operation [Formula: see text] on the group [Formula: see text]. In particular, we show that any element of [Formula: see text] is a symbol [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] is the quadratic trace form associated with the extension [Formula: see text]. As an application, we obtain certain results on the Stifel–Whitney maps [Formula: see text].

Sullivan’s construction of minimal models for topological spaces is refined for the case of a simply connected closed Riemannian manifold, $(M,\lt,\gt)$, to define a unique finitely generated field extension, $\mathbf{k}$ of $\mathbf{Q}$, baptized the harmonic field of $(M,\lt,\gt)$, and a morphism, $m:(\Lambda V,d)\rightarrow A_{DR}(M)$, from a Sullivan model defined over $\mathbf{k}$. The Sullivan model and the morphism are determined up to isomorphism, and the natural extension of $H(m)$ to $H(\Lambda V,d)\otimes_{\mathbf{k}}\mathbf{R}$ is an isomorphism; in particular, $(\Lambda V,d)$ is isomorphic to a rational Sullivan model for $M$ tensored with $\mathbf{k}$. Examples are constructed to show that every finitely generated extension field of $\mathbf{Q}$ occurs as a harmonic field of such a Riemannian manifold.

Let K K be a field and B r ( K ) Br(K) its Brauer group. If L / K L/K is a field extension, then the relative Brauer group B r ( L / K ) Br(L/K) is the kernel of the restriction map r e s L / K : B r ( K ) → B r ( L ) res_{L/K}:Br(K)\rightarrow Br(L) . A subgroup of B r ( K ) Br(K) is called an algebraic relative Brauer group if it is of the form B r ( L / K ) Br(L/K) for some algebraic extension L / K L/K . In this paper, we consider the m m -torsion subgroup B r m ( K ) Br_{m}(K) consisting of the elements of B r ( K ) Br(K) killed by m m , where m m is a positive integer, and ask whether it is an algebraic relative Brauer group. The case K = Q K=\mathbb {Q} is already interesting: the answer is yes for m m squarefree, and we do not know the answer for m m arbitrary. A counterexample is given with a two-dimensional local field K = k ( ( t ) ) K=k((t)) and m = 2 m=2 .

Let G be the separable Galois group of a finite field F of characteristic p, and X/F an imaginary hyperelliptic curve such that G acts transitively on its set W(X) of Weierstrass points. The existence of a G-invariant 2-torsion point on the Jacobian J(X) of X depends only on the parity of |W(X)|, but for large enough |F|, there exist two such curves X and X' with |W(X)|=|W(X')|, such that J(X) has (and J(X') does not have) a G-invariant 4-torsion point. The problem is equivalent to a study of the 2-,4- and 8-rank of the class number of the maximal order in the function field of such curves, and is investigated via the 2-primary class field tower. Contrary to the case of number fields, the ambiguous class depends on the discriminant, and a governing field for the 8-rank of such function fields is not known., 10 pages, LaTeX, uses `a4'

We study two invariants, ν(L/K) and λ(L/K), arising from a field extension L/K. By definition, ν(L/K) is the cardinality of the set of fields F such that K ⊆ F ⊆ L; and λ(L/K) is the supremum of the set of cardinal numbers arising as lengths of chains {Fi } of fields such that K ⊆ Fi ⊆ L. If L can be generated by one element over an infinite field K and 2 ≤ [L : K] = n < ∞, then ν(L/K) ≤ 2 n−2 + 1, with equality if L/K is Galois with the Klein four-group as Galois group. Despite positive results in the case of Abelian Galois groups, λ is not generally additive for finite-dimensional towers; moreover, maximal chains of intermediate fields can exhibit noncatenarian behavior if [L : K] = 12. The theory for infinite-dimensional algebraic extensions is strikingly different. For instance, for each infinite cardinal number N, there exists a field K of cardinality N and a chain of cardinality 2 N consisting of algebraic field extensions of K; in other words, if denotes an algebraic closure of K, then and, a forti...

The notion of R-equivalence in the set X(F) of F-points of an algebraic variety X defined over a field F was introduced by Manin in [11] and studied for linear algebraic groups by Colliot-Thelene and Sansuc in [3]. For an algebraic group G defined over a field F, the subgroup RG(F) of R-trivial elements in the group G(F) of all F-points is defined as follows. An element g belongs to RG(F) if there is a rational morphism f: Al -* G over F, defined at the points 0 and 1 such that f(O) = 1 and f(1) = g. In other words, g can be connected with the identity of the group by the image of a rational curve. The subgroup RG(F) is normal in G(F) and the factor group G(F)/RG(F) = G(F)/R is called the group of R-equivalence classes. The group of R-equivalence classes is very useful while studying the rationality problem for algebraic groups, the problem to determine whether the variety of an algebraic group is rational or stably rational. We say that a group G is R-trivial if G(E)/R = 1 for any field extension E/F. If the variety of a group G is stably rational over F, then G is R-trivial. Thus, if one can establish non-triviality of the group of R-equivalence classes G(E)/R just for one field extension E/F, the group G is not stably rational over F. The group of R-equivalence classes for adjoint semisimple classical groups was computed in [15]. This computation was used to obtain examples of non-rational adjoint algebraic groups. Consider simply connected algebraic groups of classical types. Let G = GL1 (A) be the algebraic group of invertible elements of a central simple F-algebra A of dimension n2, and let H = SL1 (A) be the subgroup in G of the reduced norm 1 elements. The group H is a simply connected group of inner type A,-1. V. VoskresenskiY [29] has shown that the natural homomorphism Ax -* K1 (A) induces an isomorphism

Soit f un revetement ramifie de P1 defini sur Q. Lorsqu'on s'interesse aux proprietes de rationalite de f sur les les corps de nombres, on peut soit exiger que la base soit P1, soit l'autoriser a etre une courbe de genre 0. Nous comparons ces deux points de vue pour les revetements non ramifies en dehors de {0,1, ∞}.

Let double-struck F be a field of characteristic ≠ 2. The determinantal complexity of a polynomial P ∈ double-struck F[x1,..., x n] is defined as the smallest size of a matrix M whose entries are linear polynomials of xi 's over double-struck F, such that P = det(M) as polynomials in double-struck F[x1,..., xn]. To determine the determinantal complexity of the permanent polynomial is a long-standing open problem. Let double-struck K be an extension field of double-struck F; then P can be viewed as a polynomial over double-struck K. We are interested in the comparison between the determinantal complexity of P over double-struck K (denoted as dcdouble-struck K(P))), and that of P over double-struck F (denoted as dcdouble-struck F(P). It is clear that dcdouble-struck K(P) ≤, dcdouble-struck F(P) and the question is whether strict inequality can happen. In this note we consider polynomials defined over ℚ. For P = x12 + ⋯ + xn2, there exists a constant multiplicative gap between dcℝ(P) and dcℂ(P): we prove dc ℝ(P) ≥ n while ⌈n/2⌉ + 1 ≥ dc ℂ(P). We also consider additive constant gaps: (1) there exists a quadratic polynomial Q ∈ ℚ [x, y], such that dcℚ-(Q) = 3 and ; (2) there exists a cubic polynomial C ∈ ℚ[x, y] with a rational zero, such that dcℚ(C) = 4 and dcℚ-(C) = 3. For additive constant gaps, geometric criteria are presented to decide when dcℚ = dcℚ-. © 2013 Springer-Verlag.

Let A be a transitive subgroup of S n . We show that the largest cyclic quotient of A has order at most n . This can be interpreted as an equivalent result about extensions of constants in the Galois closure of a covering of curves over a finite field. We also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit.

Let k ⊂ K be a field extension, where K is an algebraically closed field of any characteristic and k is the prime field. Recall the following property of Hilbert Schemes (see, for example, [1], Proposition 1.16): Suppose ⊂ × S is a flat family of closed subschemes of parametrised by a scheme S/k. Then for every closed subscheme Z ⊂ in , if [Z] denotes the Hilbert point of Z in Hilb() then the residue field of Hilb() at [Z] is the minimal field of definition for Z. Intuitively, this says that as a family parametrised by Hilb(), each fibre of lies above a point whose “co-ordinates” generate its minimal field of definition.In the following note we are concerned with a more naive form of the above situation, for which we wish to give an elementary account. Suppose ϕ(x,y) is a system of polynomial equations over k (in variables x = (x1,…, xm) and parameters y = (y1, …, yn)), such thatis a family of (possibly reducible) affine varieties in Km. Each nonempty member of this family has a unique minimal field of definition. The question arises as to whether it is possible to express this family of varieties using parameters that come, pointwise, from these minimal fields of definition. That is, is there a system of polynomial equations ψ(x, z) over k, such that each ψ(x, b) with b ∈ KN is of the form Va for some a ∈ Kn; and such that each Va ∈ is defined by ψ(x, b) for some b ∈ KN whose coordinates generate the minimal field of definition for Va? Moreover, we would like b to be obtained definably from a.

Let LF be a finite separable field extension of degree n, X a smooth quasi-projective L-scheme, and R(X) the Weil transfer of X with respect to LF. The map Z R(Z) of the set of simple cycles Z ⊂ X extends in a natural way to a map Z(X) → Z(R(X)) on the whole group of algebraic cycles Z(X). This map factors through the rational equivalence of cycles and induces this way a map of the Chow groups CH(X) → CH(R(X)), which, in its turn, produces a natural functor of the categories of Chow correspondences CV(L) → CV(F). Restricting to the graded components, one has a map Z∗(X) → Zn · ∗(R(X)), which produces a functor of the categories of degree 0 Chow correspondences CV0(L) → CV0(F), a functor of the categories of the Grothendieck Chow-motivesM(L) → M(F), as well as functors of several other classical motivic categories.

Let D be a locally compact, non-discrete and non-archimedean division algebra. Let F denote its center. Fix an integer m ≥ 1 and write I for the affine building of G = GL(m, D). Let E be a field extension of F embedded in M(m, D). Write GE and IE for the centralizers of E× in G and I respectively. Let IE denote the affine building of GE. Then there exists a unique GE-equivariant and affine map j: IE → I. It induces a bijection IE ≃ IE. In a sense to be made precise, it is compatible with the natural filtration of the parahoric subgroups.

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex1, …,xn with the following properties: The field extensionF/K(x1,…,xn) is separable and the Galois hull\(\hat F\) ofF/K(x1,…,xn) remains regular overK, i.e.K is algebraically closed in\(\hat F\). We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices iW(φk) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN, we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ℝ(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ⟂ − φ ∈ InF and give some bounds on m depending on n which assure that they have the same splitting pattern.

Letkbe any field,s = (aij)1 ≤ i, j ≤ n + 1 ∈ GLn + 1(k),K ≔ k(x1/xn + 1,···, xn/xn + 1, y1/yn + 1,···, yn/yn + 1) a field extension of transcendence degree 2noverk. The diagonal action ofsonKis defined as [Equation] [Equation] Theorem .There existz1,⋯,zn ∈ Ksuch thatK = k(x1/xn + 1,···, xn/xn + 1, z1,⋯,zn) ands(zi) = zifor 1 ≤ i ≤ n. In particular, ifk(x1/xn + 1,⋯,xn/xn + 1)〈s〉is rational ( = purely transcendental) overk, so isK〈s〉overk.

Let K be F_q((T)), or more generally any field of characteristic p equipped with a valuation having a finite residue field of q elements. Then a polynomial f(x) in K[x] having k+1 nonzero coefficients has at most q^k distinct zeros in K. We also obtain the best possible bound for the number of zeros of degree at most d over K., 6 pages, Latex 2e; to appear in Math. Res. Letters

For a prime numberpwe characterize the finitely generated maximal pro-pGalois groups of algebraic extensions of Q. This generalizes a characterization by Jensen and Prestel of the maximal abelian quotients of these Galois groups. As an application we show that the Witt rings of the algebraic extensions of Q with finitely many square classes have elementary type.

Let F be a finite field, A = F[T], and k = F(T). Let Km = k(Am) be the field extension of k obtained by adjoining the m-torsion on the Carlitz module. The class number hm of Km can be written as a product hm = h+ hThe number hm is called the relative class number. In this paper a formula for hm is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of p-th roots of unity. Some consequences of this formula are also derived. Let Q denote the rational numbers and consider the cyclotomic field Kp = Q((p) with class number hp. It is well known that this class numbers factors as a product h+hp of two integers. The number h+ is the class number of the maximal real subfield of Kp. The integer hp is called the relative class number. In [Ca] and [Ca-O] it is shown how the relative class number can be computed in terms of a certain classical determinant known as the Maillet determinant. A nice exposition of this is given in Chapter 3 of [L] (see Theorem 7.1). In this note we will give an analogue of this material in the context of cyclotomic function fields. Let F be a finite field with q elements and A = F[T] the polynomial ring over F. Let k = F(T) be the quotient field of A. For an irreducible polynomial m of degree d we will denote by Am the m-torsion on the Carlitz module and let Km = k(Am) be the "cyclotomic" function field obtained by adjoining the elements of Am to k. For the definition of the Carlitz module and its properties see [H] and [G-R]. The class number of Km, hm, factors as a product of two integers hm = h+ h-, where h+ is the class number of the "maximal real" subfield of Km, i.e. the decomposition field of the prime at infinity of k in Km, and his called the relative class number. Our aim is to give an expression for has a product of certain easily computed determinants related to the classical Maillet determinant. We begin by recalling the analytic class number formula for hwhich follows immediately from Theorem 2 of [G-R]. A character X of (A/mA)* is said to be real if its restriction to F* is the trivial character. Otherwise it is said to be non-real or imaginary. If X is imaginary, define S(W) = Za X(a), where the sum is over all monic polynomials of degree less than d. Then, (1) ]Jm = I S(W) X imaginary Received by the editors July 2, 1995 and, in revised form, November 15, 1995. 1991 Mathematics Subject Classification. Primary 11R29; Secondary 11R58, 14H05. This work was partially supported with a grant from the National Science Foundation. (?)1997 American Mathematical Society 1299 This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 04:47:14 UTC All use subject to http://about.jstor.org/terms

Let E be a field, p a prime number and F/E a finitely-generated extension of transcendency degree t. This paper shows that if the absolute Galois group GE is of nonzero cohomological p-dimension cdp(E), then the field F has Brauer p-dimension Brdp(F) ≥ t except, possibly, in case p = 2, the Sylow pro-2-subgroups of GE are of order 2, and F is a nonreal field. It announces that Brdp(F) is infinite whenever t ≥ 1 and the absolute Brauer p-dimension abrdp(E) is infinite; moreover, for each pair (m,n) of integers with 1 ≤ m ≤ n, there exists a central division F-algebra of exponent p m and Schur index p n .

Let $p$ be a prime integer and $F$ a field of characteristic 0. Let $X$ be the {\em norm variety} of a symbol in the Galois cohomology group $H^{n+1}(F,��_p^{\otimes n})$ (for some $n\geq1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F(X)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\CH(Y)\to\CH(Y_{F(X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (\dim X)/(p-1)$. One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in Appendix). Another important ingredient is {\em $A$-triviality} of $X$, the property saying that the degree homomorphism on $\CH_0(X_L)$ is injective for any field extension $L/F$ with $X(L)\ne\emptyset$. The proof involves the theory of rational correspondences reviewed in Appendix., 38 pages; final version, to appear in Ann. Sci. \'Ec. Norm. Sup\'er. (4)

Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .

The trace takes bilinear forms over a separable field extension to certain bilinear forms over the base field. This paper shows how to compute the inverse of that process. The construction implies that two known ways of classifying rational symmetric matrices under orthogonal similarity are actually the same.

Introduction. Let L be a field. An additive mapping d : L −→ L is said to be a derivation of L if d(ab) = d(a)b+ ad(b) for any a, b ∈ L. If K ⊂ L is a field extension, then a derivation d of L is said to be a K-derivation if d(αa) = αd(a) for any α ∈ K and a ∈ L. If d is a derivation of L such that d(L) = L, then we say that d is epimorphic. (a) Let K ⊂ L be a finite dimensional field extension and let d be a Kderivation of L. Then L has a direct sum decomposition L = Ker d ⊕ L′, as a K-module, where L′ is K-isomorphic to d(L). Thus dimK L > dimK d(L) yields d(L) 6= L for any K-derivation d. (b) Assume now that L is a purely transcendental extension with infinite transcendental basis {xλ; λ ∈ Λ} over Q, the field of rational numbers. Since |L| = |Λ| (the cardinalities of L and Λ), we may put L = {aλ; λ ∈ Λ}. Then the Q-derivation d of L defined by d(xλ) = aλ is epimorphic. Thus it is natural to ask whether a field extension K ⊂ L has (or has not) an epimorphic K-derivation. The purpose of this paper is to study on this problem.

Let [Formula: see text] be a group and [Formula: see text]. The 2-tuple [Formula: see text] is said to be an [Formula: see text]-Engel pair, [Formula: see text], if [Formula: see text], [Formula: see text] and [Formula: see text]. Let SL[Formula: see text] be the special linear group of degree [Formula: see text] over the field [Formula: see text]. In this paper, we show that given any field [Formula: see text], there is a field extension [Formula: see text] of [Formula: see text] with [Formula: see text] such that SL[Formula: see text] has an [Formula: see text]-Engel pair for some integer [Formula: see text]. We will also show that SL[Formula: see text] has a [Formula: see text]-Engel pair if [Formula: see text] is a field of characteristic [Formula: see text].

In this article we derive several consequences of a matricial characterization of P-recursive sequences. This characterization leads to canonical representations of these sequences. We show their uniqueness for a given sequence, up to similarity. We study their properties: operations, closed forms, d'Alembertian sequences, field extensions, positivity, extension of the sequence to $\mathbb Z$, difference Galois group.

International audience; We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $\Ba \in L[y]$, where $L=\QQ(\alpha_1, \dots, \alpha_{\ell})$ is an algebraic extension of the rational numbers. Our bounds are single exponential in $\ell$ and match the ones presented in \cite{st-issac-2011} for the case $\ell=1$. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of $\Ba$. The Boolean complexity of this approach is $\sOB(N^{4\ell+4})$, where $N$ is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of $\sOB(\min\set{N^{4\ell + 7},N^{2\ell^2+6}})$ and $\sOB( N^{2\ell+4})$, respectively. %% We implemented the algorithms in \func{C} as part of the core library of \mathematica and we illustrate their efficiency over various data sets.

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate $x^{c\log x}$. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers., Comment: 27 pages, a small change in title, final revision, to appear in Adv. Math

The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in \mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) = \infty $, $p \dagger q ^{2} - q$, for every finitely-generated transcendental extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b _{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} = b _{2}$ and $0 \le b _{p} \le a _{p} \le \infty $, equals the sequence abrd$_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, for a field $E$ of characteristic zero., Comment: LaTeX, 14 pages: published in Journal of Algebra {\bf 428} (2015), 190-204; the abstract in the Metadata updated to fit the one of the paper

Let K be a commutative ordered field and L =K(i) the quadratic extension of K with i2 = −1. Let H be the set of all Hermitian 2 × 2 matrices over the field extension (L,K) and let H(2),+ ≔ {A ∈ H ¦ det A ∈ K(2), Tr A > 0}. Then we prove that (H(2),+,⊕) is a K-loop with respect to the operation % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $$ {\rm A}\ \oplus \ {\rm B}= {1 \over {\rm TrA} + 2{\sqrt {\rm det A}}} ({\sqrt {\rm det A}}\ E +A){\rm B} ({\sqrt {\rm det A}}\ E +A) $$ where E is the identity matrix.

LetL/K be a right quadratic (skew) field extension and let be a 3-dimensional projective space overK which is embedded in a 3-dimensional projective space overL. Moreover, let ℐ be a line of which carries no point of . The main result is that — even whenL orK is a skew field — the following holds true: A Desarguesian spread of is given by the set of all lines of which are indicated by the points of ℐ. A spread of arises in this way if, and only if, there exists an isomorphism ofL onto the kernel of the spread such thatK is elementwise invariant. Furthermore, a geometric characterization of right quadratic extensions with a left degree other than 2 and of quadratic Galois extensions is given.

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L=Qα 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 OB(N10) for isolating the real roots of Bα, where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. 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 introduced by Sagraloff. For the former we prove a complexity bound of OB(N8) and for the latter a bound of OB(N7). We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.

Let K be a field, and let \(f\left( X\right) \in \mathbf{K}\left[ X \right] \) be a polynomial with coefficients in K. In general, the equation \(f\left( x\right) =0\) has solutions in an extension of K, not in K itself. For example, if K = ℚ , the equation x2 + 2 = 0 has no solution in ℚ , it has in ℂ its full set of solutions \(\left\{ i\sqrt{2},-i\sqrt{2}\right\} \), and these solutions are already in the extension \( \mathbb{Q} \left( i,\sqrt{2}\right) \subsetneq \mathbb{C} \) of ℚ . The study of algebraic field extensions is the Galois theory, recalled in Section 4.2.

we consider a situation where two processors P sub 1 and P sub 2 ar e to evaluate a collection of functions function 1,....,function 8 of two vector variables x, y, under the assumption that processor P sub 1 (respectively, P sub 2) has access only to the value of the variable x (respectively, y) and the functional form of function 1,....,function 8. We provide some new bounds on the communication complexity (the amount of information that has to be exchanged between the processors) for this problem. An almost optimal bound is derived for the case of one-way communication when the functions function 1,...., function 8 are polynomials. We also derive some new lower bounds for the case of two-way communication which improve on earlier bounds by Abelson A 80. As an application, we consider the case where x and y are n x n matrices and f(x,y) is a particular entry of the inverse of x + y. Under certain restriction on the class of allowed communication protocols, we obtain an omega(n squared) lower bound, in contrast to the omega(n) lower bound obtained by applying Abelson's results. Our results are based on certain tools from classical algebraic geometry and field extension theory.

We characterize weakly self-dual bases of the field extension\(\mathbb{F}_{q^n } \) over\(\mathbb{F}_q \), examine the existence of weakly self-dual polynomial bases, and use duality to analyze normal basis multiplication.

We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if\(\bar G\) is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of\(\bar G\) can never be maximal, and so the maximal subgroups of\(\bar G\) are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of\(\bar G\) actually gives all the maximal subgroups.