Hilbert's reciprocity law states that if " a " and " b " are in an algebraic number field containing the " n " th roots of unity then
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The famous problems of David Hilbert stimulated further development, which led to the reciprocity laws, and proofs by Teiji Takagi, Phillip Furtw�ngler, Emil Artin, Helmut Hasse and many others.
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The Artin reciprocity law, which is a high level generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field.
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The description is in terms of Frobenius elements, and generalises in a far-reaching way the quadratic reciprocity law that gives full information on the decomposition of prime numbers in quadratic fields.
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Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.
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Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.
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Reciprocity law failure is a phenomenon where the same amount of exposure ( irradiance multiplied by duration of exposure ) produces different image density when the irradiance ( and thus duration ) is varied.
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That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet " L "-function is an analytic formulation of the quadratic reciprocity law of Gauss.
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In a footnote he notes that the Eisenstein integers are the natural domain for stating and proving results on cubic reciprocity and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws.
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These laws follow easily from each version of quadratic reciprocity law stated above ( unlike with Legendre and Jacobi symbol where both the main law and the supplementary laws are needed to fully describe the quadratic reciprocity ).
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