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

(an)=(au)i=1k(api)ei

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