PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
Kronecker symbol (Definition)

The Kronecker symbol is a generalization of the Jacobi symbol to all integers.

Let $ n$ be an integer, with prime factorization $ u \cdot {p_1}^{e_1} \cdots {p_k}^{e_k}$, where $ u$ is a unit and the $ p_i$ are primes. Let $ a \geq 0$ be an integer. The Kronecker symbol $ \left(\frac{a}{n}\right)$ is defined to be

$\displaystyle \left(\frac{a}{n}\right) = \left(\frac{a}{u}\right) \prod_{i=1}^k \left(\frac{a}{p_i}\right)^{e_i} $

For odd $ p_i$, the number $ \left(\frac{a}{p_i}\right)$ is simply the usual Legendre symbol. This leaves the case when $ p_i=2$. We define $ \left(\frac{a}{2}\right)$ by

$\displaystyle \left(\frac{a}{2}\right) = \begin{cases} 0 &\text{if $a$\ is even... ...ext{if $a$\ is odd and $n \equiv 3$\ or $n \equiv 5 \pmod{8}$} \ \end{cases} $

Since it extends the Jacobi symbol, the quantity $ \left(\frac{a}{u}\right)$ is simply 1 when $ u=1$. When $ u=-1$, we define it by

$\displaystyle \left(\frac{a}{-1}\right) = \begin{cases} -1 & \text{if $a < 0$} \ 1 & \text{if $a > 0$} \ \end{cases} $

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



"Kronecker symbol" is owned by mathwizard. [ owner history (1) ]
(view preamble)

View style:

See Also: Jacobi symbol, Legendre symbol

Other names:  Kronecker-Jacobi symbol
Log in to rate this entry.
(view current ratings)

Cross-references: extensions, Legendre symbol, number, odd, primes, unit, prime factorization, integers, Jacobi symbol
There are 5 references to this entry.

This is version 3 of Kronecker symbol, born on 2004-08-24, modified 2004-08-24.
Object id is 6108, canonical name is KroneckerSymbol.
Accessed 5058 times total.

Classification:
AMS MSC: 11A07 (Number theory :: Elementary number theory :: Congruences; primitive roots; residue systems)
 11A15 (Number theory :: Elementary number theory :: Power residues, reciprocity)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)