quadratic reciprocity

English

Alternative forms

• law of quadratic reciprocity

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.

Noun

quadratic reciprocity (uncountable)

1. (number theory) 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.
   • 2007, Paul B. Garrett, Abstract Algebra‎^[1], Taylor & Francis (Chapman Hall/CRC Press), page 287:

     Yes, but we need not only Quadratic Reciprocity but also Dirichlet's theorem on primes in arithmetic progressions to see this.

   • 2009, Sam Vandervelde, Circle in a Box, American Mathematical Society, page 153:

     Gauss studied these sorts of numbers while attempting to formulate and prove higher order reciprocity laws, following his success with quadratic reciprocity.

   • 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:

     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 $F=\mathbb {Q}$.

Usage notes

There are several equivalent statements of the theorem. One version states that if p and q are odd prime numbers, $\left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}$, where $\left({p \over q}\right)$ is the Legendre symbol. This equation remains valid if $\left({p \over q}\right)$ is interpreted as a Jacobi symbol, in which case p and q are (only) required to be odd positive coprime integers. However, the value of the Jacobi symbol is less informative about whether p is a square modulo q (it can reveal that it is not, but not definitively that it it is).

The equation can be used to simplify calculation of the Legendre / Jacobi symbol.

See also

• Legendre symbol
• Jacobi symbol
• quadratic residue

Further reading

• Modular arithmetic on Wikipedia.Wikipedia
• Quadratic residue on Wikipedia.Wikipedia