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- Abhyankar's_conjecture abstract "In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p. This problem was solved in 1994 by work of Michel Raynaud and David Harbater.The problem involves a finite group G, a prime number p, and the function field of nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p.The question addresses the existence of Galois extensions L of K(C), with G as Galois group, and with restricted ramification. From a geometric point of view L corresponds to another curve C′, and a morphismπ : C′ → C.Ramification geometrically, and by analogy with the case of Riemann surfaces, consists of a finite set S of points x on C, such that π restricted to the complement of S in C is an étale morphism. In Abhyankar's conjecture, S is fixed, and the question is what G can be. This is therefore a special type of inverse Galois problem.The subgroup p(G) is defined to be the subgroup generated by all the Sylow subgroups of G for the prime number p. This is a normal subgroup, and the parameter n is defined as the minimum number of generators ofG/p(G).Then for the case of C the projective line over K, the conjecture states that G can be realised as a Galois group of L, unramified outside S containing s + 1 points, if and only ifn ≤ s.This was proved by Raynaud.For the general case, proved by Harbater, let g be the genus of C. Then G can be realised if and only ifn ≤ s + 2 g.↑ ↑ ↑".
- Abhyankar's_conjecture wikiPageExternalLink conjecture.html.
- Abhyankar's_conjecture wikiPageID "3832555".
- Abhyankar's_conjecture wikiPageRevisionID "487498375".
- Abhyankar's_conjecture hasPhotoCollection Abhyankar's_conjecture.
- Abhyankar's_conjecture title "Abhyankar's conjecture".
- Abhyankar's_conjecture urlname "AbhyankarsConjecture".
- Abhyankar's_conjecture subject Category:Algebraic_curves.
- Abhyankar's_conjecture subject Category:Conjectures.
- Abhyankar's_conjecture subject Category:Galois_theory.
- Abhyankar's_conjecture subject Category:Theorems_in_abstract_algebra.
- Abhyankar's_conjecture type Abstraction100002137.
- Abhyankar's_conjecture type AlgebraicCurves.
- Abhyankar's_conjecture type Attribute100024264.
- Abhyankar's_conjecture type Cognition100023271.
- Abhyankar's_conjecture type Communication100033020.
- Abhyankar's_conjecture type Concept105835747.
- Abhyankar's_conjecture type Conjectures.
- Abhyankar's_conjecture type Content105809192.
- Abhyankar's_conjecture type Curve113867641.
- Abhyankar's_conjecture type Hypothesis105888929.
- Abhyankar's_conjecture type Idea105833840.
- Abhyankar's_conjecture type Line113863771.
- Abhyankar's_conjecture type Message106598915.
- Abhyankar's_conjecture type Proposition106750804.
- Abhyankar's_conjecture type PsychologicalFeature100023100.
- Abhyankar's_conjecture type Shape100027807.
- Abhyankar's_conjecture type Speculation105891783.
- Abhyankar's_conjecture type Statement106722453.
- Abhyankar's_conjecture type Theorem106752293.
- Abhyankar's_conjecture type TheoremsInAbstractAlgebra.
- Abhyankar's_conjecture comment "In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p.".
- Abhyankar's_conjecture label "Abhyankar's conjecture".
- Abhyankar's_conjecture sameAs m.0b2b29.
- Abhyankar's_conjecture sameAs Q4667492.
- Abhyankar's_conjecture sameAs Q4667492.
- Abhyankar's_conjecture sameAs Abhyankar's_conjecture.
- Abhyankar's_conjecture wasDerivedFrom Abhyankar's_conjecture?oldid=487498375.
- Abhyankar's_conjecture isPrimaryTopicOf Abhyankar's_conjecture.