Math 2.0

Number theory Quadratic residues and reciprocity Theorem

Law of quadratic reciprocity

Builds on Legendre symbolFirst supplement to quadratic reciprocitySecond supplement to quadratic reciprocity

Statement

Let pp and qq be distinct odd primes. Then (pq)(qp)=(−1)p−12⋅q−12,\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}}, where the symbols are Legendre symbols.

Remarks

So (pq)=(qp)\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right) unless p≡q≡3(mod4)p \equiv q \equiv 3 \pmod 4. In that case (pq)=−(qp)\left(\frac{p}{q}\right) = -\left(\frac{q}{p}\right). For example, (73)=(13)=1\left(\frac{7}{3}\right) = \left(\frac{1}{3}\right) = 1, and 3≡7≡3(mod4)3 \equiv 7 \equiv 3 \pmod 4. So (37)=−1\left(\frac{3}{7}\right) = -1. With the first supplement, the second supplement and multiplicativity, the law computes every Legendre symbol.

History

Euler and Legendre conjectured the law. Gauss first proved it in 1796, and he published proofs in his Disquisitiones Arithmeticae of 1801. Gauss called it the fundamental theorem.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
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…statementLegendre symbol, definition, no evidence levelLegendre symboldefinitionSecond supplement to quadratic reciprocity, statement, no evidence levelSecond supplement to …statementEuler's criterion, statement, no evidence levelEuler's criterionstatementFirst supplement to quadratic reciprocity, statement, no evidence levelFirst supplement to q…statementLaw of quadratic reciprocity, statement, no evidence levelLaw of quadratic reci…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. Legendre symbol definition
  7. Second supplement to quadratic reciprocity statement
  8. Euler's criterion statement
  9. First supplement to quadratic reciprocity statement
  10. Law of quadratic reciprocity statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quadratic_reciprocity, Statement of the theorem
  2. Web https://en.wikipedia.org/wiki/Quadratic_reciprocity, History and alternative statements
About this page
Tag085MJS6907
URIm2://085MJS6907
Aliasesquadratic-reciprocity
Kindstatement, theorem
WritingW1 stated
MSC11A15
StatusThis copy of the site has no build data. Run m2 status 085MJS6907.
UsesLegendre symbol, First supplement to quadratic reciprocity, Second supplement to quadratic reciprocity
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by