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1.
This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types. 相似文献
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Przemysław Koprowski 《Czechoslovak Mathematical Journal》2007,57(4):1311-1319
We present criteria for a pair of maps to constitute a quaternion-symbol equivalence (or a Hilbert-symbol equivalence if we
deal with global function fields) expressed in terms of vanishing of the Clifford invariant. In principle, we prove that a
local condition of a quaternion-symbol equivalence can be transcribed from the Brauer group to the Brauer-Wall group. 相似文献
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Witt kernels of quadratic forms for purely inseparable multiquadratic extensions in characteristic 2
Ahmed Laghribi 《Proceedings of the American Mathematical Society》2006,134(9):2481-2486
The aim of this paper is to give a complete answer to the question of hyperbolicity of nonsingular quadratic forms over purely inseparable multiquadratic extensions in characteristic . This completes partial computations of Mammone and Moresi.
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Przemysław Koprowski 《Mathematische Zeitschrift》2002,242(2):323-345
We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there
are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones.
The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation
in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology
of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically
connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the
Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field.
Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002 相似文献
7.
Jonathan Shick 《代数通讯》2013,41(4):1371-1388
The local cohomology modules HJ I(M) of a Matlis reflexive module are shown to be I-cofinite when j >= 1 and have finite Bass numbers when j >= 0, where I is an ideal satisfying any one of a list of properties. In addition, we show that the completion of a Matlis reflexive module is finitely generated over the completion of the ring and we classify Matlis reflexive modules over a one dimensional ring. 相似文献
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B. M. Kim 《Commentarii Mathematici Helvetici》2000,75(3):410-414
In this paper, we will prove there are infinitely many integers n such that n
2— 1 is square-free and admits universal octonary diagonal quadratic forms.
Received: November 2, 1998. 相似文献
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Arpan Dutta 《Mathematische Nachrichten》2023,296(10):4745-4768
Given a valued field and its completion , we study the set of all possible extensions of v to . We show that any such extension is closely connected with the underlying subextension . The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of . We also give necessary and sufficient conditions for to be dense in . 相似文献
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Detlev W. Hoffmann 《Transactions of the American Mathematical Society》1996,348(8):3267-3281
Let and be anisotropic quadratic forms over a field of characteristic not . Their function fields and are said to be equivalent (over ) if and are isotropic. We consider the case where and is divisible by an -fold Pfister form. We determine those forms for which becomes isotropic over if , and provide partial results for . These results imply that if and are equivalent and , then is similar to over . This together with already known results yields that if is of height and degree or , and if , then and are equivalent iff and are isomorphic over .
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V. Suresh 《Journal of Number Theory》2004,107(1):80-94
Let k be a field of characteristic not equal to 2. For n≥1, let denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , there exist such that αi=(a)∪(bi), where for any λ∈k∗, (λ) denotes the image of k∗ in . In this paper we prove a higher dimensional analogue of the Tate's lemma. 相似文献
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Let K be a complete discretely valued field with residue field κ . If char(K)=0, char(κ)=2 and [κ:κ2]=d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable over K is bounded by N. The method of proof is via introducing the notion of uniform boundedness for the p-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to the finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over K is uniformly bounded. 相似文献
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We complete a classification of quadratic forms over a field of characteristic 2 of type (1,3) that become isotropic over the function field of a quadric. 相似文献
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Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in : where is the Legendre symbol.
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In this paper, we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field with characteristic other than 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. The calculation is then completed using classical results for bilinear spaces over fields. 相似文献
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Lucyna Szczepanik 《代数通讯》2013,41(9):3359-3367
ABSTRACT In this note it is proved that certain level sets of some real proper polynomial maps are nothing but spheres. As an application of this, we provide new proofs of Theorems 1.1, 1.2 and of the fundamental theorem of algebra. In addition, we show that every strictly convex (concave) polynomial map is proper. The latter implies that every real polynomial map g(x): R n → R n , whose Jacobian matrix is symmetric and has nonzero eigenvalues of the same sign, is a homeomorphism of R n onto R n . 相似文献
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In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.
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Zhaoxi Wang 《分析论及其应用》2009,25(1):92-100
The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho¨lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a “chain of function spaces“ over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields. 相似文献