Kronecker symbol

Let n be an integer, with prime factorizationMathworldPlanetmath up1e1pkek, where u is a unit and the pi are primes. Let a0 be an integer. The Kronecker symbol (an) is defined to be


For odd pi, the number (api) is simply the usual Legendre symbolMathworldPlanetmath. This leaves the case when pi=2. We define (a2) by

(a2)={0if a is even1if a is odd and n1 or n7(mod8)-1if a is odd and n3 or n5(mod8)

Since it extends the Jacobi symbol, the quantity (au) is simply 1 when u=1. When u=-1, we define it by

(a-1)={-1if a<01if a>0

These extensionsPlanetmathPlanetmath suffice to define the Kronecker symbol for all integer values n.

Title Kronecker symbol
Canonical name KroneckerSymbol
Date of creation 2013-03-22 14:33:21
Last modified on 2013-03-22 14:33:21
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 6
Author mathwizard (128)
Entry type Definition
Classification msc 11A07
Classification msc 11A15
Synonym Kronecker-Jacobi symbol
Related topic JacobiSymbol
Related topic LegendreSymbol