On 8 April he became the first to prove the quadratic reciprocity law.
2.
Its immense bibliography includes literature citations for 196 different published proofs for the quadratic reciprocity law.
3.
The latter property is called the " global reciprocity law " and is a far reaching generalization of the Gauss quadratic reciprocity law.
4.
In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss; in connection to this, the Legendre symbol is named after him.
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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.
8.
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 ).
9.
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.
10.
Just as the quadratic reciprocity law for the Legendre symbol is also true for the Jacobi symbol, the requirement that the numbers be prime is not needed; it suffices that they be odd relatively prime nonunits.
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