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In number theory, the Kronecker symbol, written as or , is a generalization of the Jacobi symbol to all integers . It was introduced by Leopold Kronecker (1885, page 770).

Contents

DefinitionEdit

Let   be a non-zero integer, with prime factorization

 

where   is a unit (i.e.,  ), and the   are primes. Let   be an integer. The Kronecker symbol   is defined by

 

For odd  , the number   is simply the usual Legendre symbol. This leaves the case when  . We define   by

 

Since it extends the Jacobi symbol, the quantity   is simply   when  . When  , we define it by

 

Finally, we put

 

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

Some authors only define the Kronecker symbol for more restricted values; for example,   congruent to   and  .

Table of valuesEdit

The following is a table of values of Kronecker symbol   with n, k ≤ 30.

k
n
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0
3 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0
4 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
5 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0
6 1 0 0 0 1 0 1 0 0 0 1 0 −1 0 0 0 −1 0 −1 0 0 0 −1 0 1 0 0 0 1 0
7 1 1 −1 1 −1 −1 0 1 1 −1 1 −1 −1 0 1 1 −1 1 −1 −1 0 1 1 −1 1 −1 −1 0 1 1
8 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0
9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0
10 1 0 1 0 0 0 −1 0 1 0 −1 0 1 0 0 0 −1 0 −1 0 −1 0 −1 0 0 0 1 0 −1 0
11 1 −1 1 1 1 −1 −1 −1 1 −1 0 1 −1 1 1 1 −1 −1 −1 1 −1 0 1 −1 1 1 1 −1 −1 −1
12 1 0 0 0 −1 0 1 0 0 0 −1 0 1 0 0 0 −1 0 1 0 0 0 −1 0 1 0 0 0 −1 0
13 1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 0 1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 0 1 −1 1 1
14 1 0 1 0 1 0 0 0 1 0 −1 0 1 0 1 0 −1 0 1 0 0 0 1 0 1 0 1 0 −1 0
15 1 1 0 1 0 0 −1 1 0 0 −1 0 −1 −1 0 1 1 0 1 0 0 −1 1 0 0 −1 0 −1 −1 0
16 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
17 1 1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1 −1 1 1 0 1 1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1
18 1 0 0 0 −1 0 1 0 0 0 −1 0 −1 0 0 0 1 0 −1 0 0 0 1 0 1 0 0 0 −1 0
19 1 −1 −1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 1 1 −1 0 1 −1 −1 1 1 1 1 −1 1 −1 1
20 1 0 −1 0 0 0 −1 0 1 0 1 0 −1 0 0 0 −1 0 1 0 1 0 −1 0 0 0 −1 0 1 0
21 1 −1 0 1 1 0 0 −1 0 −1 −1 0 −1 0 0 1 1 0 −1 1 0 1 −1 0 1 1 0 0 −1 0
22 1 0 −1 0 −1 0 −1 0 1 0 0 0 1 0 1 0 −1 0 1 0 1 0 1 0 1 0 −1 0 1 0
23 1 1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 −1 −1 −1 0 1 1 1 1 −1 1 −1
24 1 0 0 0 1 0 1 0 0 0 1 0 −1 0 0 0 −1 0 −1 0 0 0 −1 0 1 0 0 0 1 0
25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
26 1 0 −1 0 1 0 −1 0 1 0 1 0 0 0 −1 0 1 0 1 0 1 0 1 0 1 0 −1 0 −1 0
27 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0
28 1 0 −1 0 −1 0 0 0 1 0 1 0 −1 0 1 0 −1 0 −1 0 0 0 1 0 1 0 −1 0 1 0
29 1 −1 −1 1 1 1 1 −1 1 −1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 1 −1 −1 1 0 1
30 1 0 0 0 0 0 −1 0 0 0 1 0 1 0 0 0 1 0 −1 0 0 0 1 0 0 0 0 0 1 0

PropertiesEdit

The Kronecker symbol shares many basic properties of the Jacobi symbol, under certain restrictions:

  •   if  , otherwise  .
  •   unless  , one of   is zero and the other one is negative.
  •   unless  , one of   is zero and the other one has odd part (definition below) congruent to  .
  • For  , we have   whenever   If additionally   have the same sign, the same also holds for  .
  • For  ,  , we have   whenever  

On the other hand, the Kronecker symbol does not have the same connection to quadratic residues as the Jacobi symbol. In particular, the Kronecker symbol   for even   can take values independently on whether   is a quadratic residue or nonresidue modulo  .

Quadratic reciprocityEdit

The Kronecker symbol also satisfies the following versions of quadratic reciprocity law.

For any nonzero integer  , let   denote its odd part:   where   is odd (for  , we put  ). Then the following symmetric version of quadratic reciprocity holds for every pair of integers   such that  :

 

where the   sign is equal to   if   or   and is equal to   if   and  .

There is also equivalent non-symmetric version of quadratic reciprocity that holds for every pair of relatively prime integers  :

 

For any integer   let  . Then we have another equivalent non-symmetric version that states

 

for every pair of integers   (not necessarily relatively prime).

The supplementary laws generalize to the Kronecker symbol as well. These laws follow easily from each version of quadratic reciprocity law stated above (unlike with Legendre and Jacobi symbol where both the main law and the supplementary laws are needed to fully describe the quadratic reciprocity).

For any integer   we have

 

and for any odd integer   it's

 

Connection to Dirichlet charactersEdit

If   and  , the map   is a real Dirichlet character of modulus   Conversely, every real Dirichlet character can be written in this form with   (for   it's  ).

In particular, primitive real Dirichlet characters   are in a 1–1 correspondence with quadratic fields  , where   is a nonzero square-free integer (we can include the case   to represent the principal character, even though it is not a proper quadratic field). The character   can be recovered from the field as the Artin symbol  : that is, for a positive prime  , the value of   depends on the behaviour of the ideal   in the ring of integers  :

 

Then   equals the Kronecker symbol  , where

 

is the discriminant of  . The conductor of   is  .

Similarly, if  , the map   is a real Dirichlet character of modulus   However, not all real characters can be represented in this way, for example the character   cannot be written as   for any  . By the law of quadratic reciprocity, we have  . A character   can be represented as   if and only if its odd part  , in which case we can take  .

ReferencesEdit

  • Kronecker, L. (1885), "Zur Theorie der elliptischen Funktionen", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 761–784
  • Montgomery, Hugh L; Vaughan, Robert C. (2007). Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics. 97. Cambridge University Press . ISBN 0-521-84903-9. Zbl 1142.11001.

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