Math 2.0

Number theory Theorem

First supplement to quadratic reciprocity

Builds on Legendre symbolEuler's criterion

Statement

Let pp be an odd prime. Then (−1p)=(−1)(p−1)/2,\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2}, where the left side is a Legendre symbol.

Remarks

So (−1p)=1\left(\frac{-1}{p}\right) = 1 if p≡1(mod4)p \equiv 1 \pmod 4, and (−1p)=−1\left(\frac{-1}{p}\right) = -1 if p≡3(mod4)p \equiv 3 \pmod 4. The proof is Euler's criterion with a=−1a = -1. For example, 22=4≡−1(mod5)2^2 = 4 \equiv -1 \pmod 5, but −1-1 is not a square modulo 77.

Reading path

To understand this page from the ground up, read these 8 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…statementLegendre symbol, definition, no evidence levelLegendre symboldefinitionEuler's criterion, statement, no evidence levelEuler's criterionstatementFirst supplement to quadratic reciprocity, statement, no evidence levelFirst supplement to q…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. Euler's criterion statement
  8. First supplement to quadratic reciprocity statement

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quadratic_reciprocity, q = ±1 and the first supplement
About this page
Tag0PHBS78HKT
URIm2://0PHBS78HKT
Kindstatement, theorem
WritingW0 stub
MSC11A15
StatusThis copy of the site has no build data. Run m2 status 0PHBS78HKT.
UsesLegendre symbol, Euler's criterion

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