Kronecker symbol
Encyclopedia
In number theory
, the Kronecker symbol, written as
or (a|n), is a generalization of the Jacobi symbol
to all integer
s n. It was introduced by Leopold Kronecker
.
where u is a unit
(i.e., u is 1 or −1), and the pi are prime
s. Let a be an integer. The Kronecker symbol (a|n) is defined by
For odd pi, the number (a|pi) is simply the usual Legendre symbol
. This leaves the case when pi = 2. We define (a|2) by
Since it extends the Jacobi symbol, the quantity (a|u) is simply 1 when u = 1. When u = −1, we define it by
Finally, we put
These extensions suffice to define the Kronecker symbol for all integer values n.
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Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Kronecker symbol, written as
or (a|n), is a generalization of the Jacobi symbolJacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
to all integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s n. It was introduced by Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
.
Definition
Let n be a non-zero integer, with prime factorizationwhere u is a unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
(i.e., u is 1 or −1), and the pi are prime
Prime
A prime is a natural number that has exactly two distinct natural number divisors: 1 and itself.Prime or PRIME may also refer to:In mathematics:*Prime , the ′ mark, typically used as a suffix...
s. Let a be an integer. The Kronecker symbol (a|n) is defined by
For odd pi, the number (a|pi) is simply the usual Legendre symbol
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
. This leaves the case when pi = 2. We define (a|2) by
Since it extends the Jacobi symbol, the quantity (a|u) is simply 1 when u = 1. When u = −1, we define it by
Finally, we put
These extensions suffice to define the Kronecker symbol for all integer values n.
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