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Local Symbol (other)
In mathematics, local symbol may refer to: * The local Artin symbol in Artin reciprocity * The local symbol used to formulate Weil reciprocity In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field ''K''(''C'') of an algebraic curve ''C'' over an algebraically closed field ''K''. Given functions ''f'' and ''g'' in ''K''(''C''), i.e. rational func ... * A Steinberg symbol on a local field {{Mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Artin Symbol
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem. Statement Let L/K be a Galois extension of global fields and C_L stand for the idèle class group of L. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol mapNeukirch (1999) p.391 : \theta: C_K/ \to \operatorname(L/K)^, where \text denotes the abelianization of a group. The map \theta is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Reciprocity
In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field ''K''(''C'') of an algebraic curve ''C'' over an algebraically closed field ''K''. Given functions ''f'' and ''g'' in ''K''(''C''), i.e. rational functions on ''C'', then :''f''((''g'')) = ''g''((''f'')) where the notation has this meaning: (''h'') is the divisor of the function ''h'', or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of ''f'' and ''g'' have disjoint support (which can be removed). In the case of the projective line, this can be proved by manipulations with the resultant of polynomials. To remove the condition of disjoint support, for each point ''P'' on ''C'' a ''local symbol'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |