Math 2.0

Number theory Theorem

Quadratic reciprocity for the Jacobi symbol

Builds on Jacobi symbolLaw of quadratic reciprocity

Statement

Let mm and nn be coprime positive odd integers. Then (mn)(nm)=(−1)m−12⋅n−12,\left(\frac{m}{n}\right) \left(\frac{n}{m}\right) = (-1)^{\frac{m-1}{2} \cdot \frac{n-1}{2}}, where the symbols are Jacobi symbols.

Remarks

The supplements also extend: for a positive odd nn, (−1n)=(−1)(n−1)/2\left(\frac{-1}{n}\right) = (-1)^{(n-1)/2} and (2n)=(−1)(n2−1)/8\left(\frac{2}{n}\right) = (-1)^{(n^2-1)/8}. The proof uses the law of quadratic reciprocity for each pair of prime factors. With this law, one computes (an)\left(\frac{a}{n}\right) by an algorithm like the Euclidean algorithm, with no factorization.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionPrime number, definition, no evidence levelPrime numberdefinitionQuadratic residue, definition, no evidence levelQuadratic residuedefinitionFermat's little theorem, statement, no evidence levelFermat's little theor…statementPrime factorization, definition, no evidence levelPrime factorizationdefinitionLegendre symbol, definition, no evidence levelLegendre symboldefinitionSecond supplement to quadratic reciprocity, statement, no evidence levelSecond supplement to …statementEuler's criterion, statement, no evidence levelEuler's criterionstatementJacobi symbol, definition, no evidence levelJacobi symboldefinitionFirst supplement to quadratic reciprocity, statement, no evidence levelFirst supplement to q…statementLaw of quadratic reciprocity, statement, no evidence levelLaw of quadratic reci…statementQuadratic reciprocity for the Jacobi symbol, statement, no evidence levelQuadratic reciprocity…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Congruence modulo n definition
  3. Prime number definition
  4. Quadratic residue definition
  5. Fermat's little theorem statement
  6. Prime factorization definition
  7. Legendre symbol definition
  8. Second supplement to quadratic reciprocity statement
  9. Euler's criterion statement
  10. Jacobi symbol definition
  11. First supplement to quadratic reciprocity statement
  12. Law of quadratic reciprocity statement
  13. Quadratic reciprocity for the Jacobi symbol statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Jacobi_symbol, Properties
About this page
Tag0YY06KP3ZQ
URIm2://0YY06KP3ZQ
Kindstatement, theorem
WritingW0 stub
MSC11A15
StatusThis copy of the site has no build data. Run m2 status 0YY06KP3ZQ.
UsesJacobi symbol, Law of quadratic reciprocity

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