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Kronecker Symbol

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The Kronecker symbol is an extension of the Jacobi symbol (n/m) to all integers. It is variously written as (n/m) or (n/m) (Cohn 1980; Weiss 1998, p. 236) or (n|m) (Dickson 2005). The Kronecker symbol can be computed using the normal rules for the Jacobi symbol

((ab)/(cd))=(a/(cd))(b/(cd))
(1)
=((ab)/c)((ab)/d)
(2)
=(a/c)(b/c)(a/d)(b/d)
(3)

plus additional rules for m=-1,

(n/-1)={-1 for n<0; 1 for n>0,
(4)

and m=2. The definition for (n/2) is variously written as

(n/2)={0 for n even; 1 for n odd, n=+/-1 (mod 8); -1 for n odd, n=+/-3 (mod 8)
(5)

or

(n/2)={0 for 4|n; 1 for n=1 (mod 8); -1 for n=5 (mod 8); undefined otherwise
(6)

(Cohn 1980). Cohn's form "undefines" (n/2) for singly even numbers n=2 (mod 4) and n=-1,3 (mod 8), probably because no other values are needed in applications of the symbol involving the binary quadratic form discriminants d of quadratic fields, where m>0 and d always satisfies d=0,1 (mod 4).

The Kronecker symbol is implemented in the Wolfram Language as KroneckerSymbol[n, m].

The Kronecker symbol (d/n) is a real number theoretic character modulo d, and is, in fact, essentially the only type of real primitive character (Ayoub 1963).

KroneckerSymbol

The illustration above and table below summarize (k/n) for n=1, 2, ... and small |k|.

kOEISperiod(k/1),(k/2),(k/3),...
-6A109017241, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...
-501, -1, 1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, -1, 0, 1, -1, -1, -1, 0, ...
-441, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...
-331, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, ...
-281, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, ...
-1A0349471, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, ...
01, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
111, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2A09133781, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, ...
3A0913381, -1, 0, 1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, ...
4A00003521, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
5A08089151, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, ...
6241, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, ...

For values of d corresponding to primitive Dirichlet L-series L_d(s), the period of (d/n) equals d. For d=-1, -2, ..., the periods of (d/n) are 0, 8, 3, 4, 0, 24, 7, 8, 0, 40, 11, 6, ... (OEIS A117888) and for d=1, 2, ... they are 1, 8, 0, 2, 5, 24, 0, 8, 3, 40, 0, 12, ... (OEIS A117889). Here, 0 indicates that the sequence is not periodic.

See also

Class Number, Dirichlet L-Series, Jacobi Symbol, Legendre Symbol, Number Theoretic Character, Primitive Character, Quadratic Residue

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References

Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.Cohn, H. Advanced Number Theory. New York: Dover, p. 35, 1980.Dickson, L. E. "Kronecker's Symbol." §48 in Introduction to the Theory of Numbers. New York: Dover, p. 77, 1957.Sloane, N. J. A. Sequences A000035/M0001, A034947, A080891, A091337, A091338, A109017, A117888, and A117889 in "The On-Line Encyclopedia of Integer Sequences."Weiss, E. Algebraic Number Theory. New York: Dover, 1998.

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Kronecker Symbol

Cite this as:

Weisstein, Eric W. "Kronecker Symbol." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KroneckerSymbol.html

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