“Quadratic Reciprocity Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Quadratic Reciprocity Theorem

释义

Quadratic Reciprocity Theorem

Also called the Gauß. If and are distinctOdd Primes, then the Congruences

are both solvable or both unsolvable unless both

and

leave the remainder 3 when divided by 4 (in which case oneof the Congruences is solvable and the other is not). Written symbolically,

where

is known as a Legendre

was the first to publish a proof, but it was fallacious.Gauß

was the first to publish a correct proof. The quadratic reciprocity theorem was Gauss's favorite theorem fromNumber Theory, and he devised many proofs of it over his lifetime.

See also Jacobi Symbol, Kronecker Symbol, Legendre Symbol, Quadratic Residue, Reciprocity Theorem

References

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 39, 1996.

Ireland, K. and Rosen, M. ``Quadratic Reciprocity.'' Ch. 5 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 50-65, 1990.

Nagell, T. ``Theory of Quadratic Residues.'' Ch. 4 in Introduction to Number Theory. New York: Wiley, 1951.

Riesel, H. ``The Law of Quadratic Reciprocity.'' Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 279-281, 1994.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 42-49, 1993.

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