Dictionary

quadratic reciprocity en · NOUN

Etymology

The theorem highlights a particular form of reciprocity in the solvability of the quadratic equation a² = b in modular arithmetic. It was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss.

Meanings

1. (uncountable) The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.
   • Yes, but we need not only Quadratic Reciprocity but also Dirichlet's theorem on primes in arithmetic progressions to see this. 2007, Paul B. Garrett, Abstract Algebra, Taylor & Francis (Chapman Hall/CRC Press), page 287:
   • Gauss studied these sorts of numbers while attempting to formulate and prove higher order reciprocity laws, following his success with quadratic reciprocity. 2009, Sam Vandervelde, Circle in a Box, American Mathematical Society, page 153:
   • An interesting consequence of the above algorithm is that one can evaluate the Jacobi symbol in deterministic polynomial time in certain cases analogous to the way (“reduce and flip”) that one computes this symbol using quadratic reciprocity in the case F#61;#92;mathbb#123;Q#125;. 2013, J. Voight, “Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms”, in Krishnaswami Alladi, Manjul Bhargava, David Savitt, Pham Huu Tiep, editors, Quadratic and Higher Degree Forms, Springer, page 284:

Forms

Relateds

quadratic residue · Jacobi symbol · Legendre symbol

wikipedia: Adrien-Marie Legendre · wikipedia: Carl Friedrich Gauss · wikipedia: Leonhard Euler