共查询到20条相似文献,搜索用时 31 毫秒
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Anuj Jakhar Sudesh K. Khanduja Neeraj Sangwan 《Journal of Pure and Applied Algebra》2018,222(4):889-899
Let v be a Krull valuation of a field with valuation ring . Let θ be a root of an irreducible trinomial belonging to . In this paper, we give necessary and sufficient conditions involving only for to be integrally closed. In the particular case when v is the p-adic valuation of the field of rational numbers, and , then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup in , where is the ring of algebraic integers of K. As an application, it is deduced that for any algebraic number field K and any quadratic field L not contained in K, we have if and only if the discriminants of K and L are coprime. 相似文献
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Bruce Olberding 《Journal of Pure and Applied Algebra》2018,222(8):2267-2287
Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let and , where denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring . We show that if and is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that . If any countable subset of points is removed from Y, then the resulting set remains a representation of . Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring with specific focus on topological conditions that guarantee is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques. 相似文献
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Gábor Korchmáros Maria Montanucci Pietro Speziali 《Journal of Pure and Applied Algebra》2018,222(7):1810-1826
Let be the algebraic closure of a finite field of odd characteristic p. For a positive integer m prime to p, let be the transcendence degree 1 function field defined by . Let and . The extension is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus , p-rank (Hasse–Witt invariant) and a -automorphism group of order at least . In this paper we prove that this subgroup is the full -automorphism group of K; more precisely where Δ is an elementary abelian p-group of order and D has an index 2 cyclic subgroup of order . In particular, , and if K is ordinary (i.e. ) then . On the other hand, if G is a solvable subgroup of the -automorphism group of an ordinary, transcendence degree 1 function field L of genus defined over , then ; see [15]. This shows that K hits this bound up to the constant .Since has several subgroups, the fixed subfield of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in is large enough. This possibility is worked out for subgroups of Δ. 相似文献
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We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in . Therefore “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups. 相似文献
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Shi-Chao Chen 《Comptes Rendus Mathematique》2018,356(11-12):1081-1084
Let and be the coefficients of the Rogers–Ramanujan identities. We obtain asymptotic formulas for the number of odd values of for odd n, and for even n, which improve Gordon's results. We also obtain lower bounds for the number of odd values of for even n, and for odd n. 相似文献
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Saïd Asserda 《Comptes Rendus Mathematique》2006,342(6):393-398
Let be a complete Riemannian manifold without boundary of dimension n and V be a vector field on M such that is bounded. Suppose that outside a compact set of M, where denotes the upper eigenvalue of ?V and are non-negative decreasing functions such that . There exists positive numbers and which depend only on n and such that if h is a function defined on M with and , where , where is a sequence of M such that , then the equation has no positive solution on M. To cite this article: S. Asserda, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 相似文献
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The volume distance from a point p to a convex hypersurface is defined as the minimum -volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane we obtain, after normalization, a symmetric bilinear form Q.In this paper, we prove that Q converges to the affine Blaschke metric when we approximate the hypersurface along a curve whose points are centroids of parallel sections. We also show that the rate of this convergence is given by a bilinear form associated with the shape operator of M. These convergence results provide a geometric interpretation of the Blaschke metric and the shape operator in terms of the volume distance. 相似文献
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《Advances in Applied Mathematics》2009,42(4):510-529
We consider the situation that M and N are 3-connected matroids such that and is a cocircuit of M with the property that has an N-minor for some . We show that either there is an element such that or is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of and an element x such that is 3-connected with an N-minor. 相似文献
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