Kronecker symbol

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This article is about the symbol in number theory. For other uses, see Kronecker delta.

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

Definition[edit]

Let n be a non-zero integer, with prime factorization

n=u \cdot p_1^{e_1} \cdots p_k^{e_k},

where u is a unit (i.e., u=\pm1), and the p_i are primes. Let a be an integer. The Kronecker symbol (a|n) is defined by

 \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 (a|p_i) is simply the usual Legendre symbol. This leaves the case when p_i=2. We define (a|2) by

 \left(\frac{a}{2}\right) = 
\begin{cases}
 0 & \mbox{if }a\mbox{ is even,} \\
 1 & \mbox{if } a \equiv \pm1 \pmod{8},  \\
-1 & \mbox{if } a \equiv \pm3 \pmod{8}.
\end{cases}

Since it extends the Jacobi symbol, the quantity (a|u) is simply 1 when u=1. When u=-1, we define it by

 \left(\frac{a}{-1}\right) = \begin{cases} -1 & \mbox{if }a < 0, \\ 1 & \mbox{if } a \ge 0. \end{cases}

Finally, we put

\left(\frac a0\right)=\begin{cases}1&\text{if }a=\pm1,\\0&\text{otherwise.}\end{cases}

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

Some authors only define the Kronecker symbol for more restricted values; for example, a congruent to 0,1\bmod4 and n>0.

Table of values[edit]

The following is a table of values of Kronecker symbol \left(\frac{k}{n}\right) with n, k ≤ 30.

n \ k 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

Properties[edit]

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

  • \left(\tfrac an\right)=\pm1 if \gcd(a,n)=1, otherwise \left(\tfrac an\right)=0.
  • \left(\tfrac{ab}n\right)=\left(\tfrac an\right)\left(\tfrac bn\right) unless n=-1, one of a,b is zero and the other one is negative.
  • \left(\tfrac a{mn}\right)=\left(\tfrac am\right)\left(\tfrac an\right) unless a=-1, one of m,n is zero and the other one has odd part (definition below) congruent to 3\bmod4.
  • For n>0, we have \left(\tfrac an\right)=\left(\tfrac bn\right) whenever a\equiv b\bmod\begin{cases}4n,&n\equiv2\pmod 4,\\n&\text{otherwise.}\end{cases} If additionally a,b have the same sign, the same also holds for n<0.
  • For a\not\equiv3\pmod4, a\ne0, we have \left(\tfrac am\right)=\left(\tfrac an\right) whenever m\equiv n\bmod\begin{cases}4|a|,&a\equiv2\pmod 4,\\|a|&\text{otherwise.}\end{cases}

Quadratic reciprocity[edit]

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

For any nonzero integer n, let n' denote its odd part: n=2^en' where n' is odd (for n=0, we put 0'=1). Then the following symmetric version of quadratic reciprocity holds for every pair of integers m,n such that \gcd(m,n)=1:

\left(\frac mn\right)\left(\frac nm\right)=\pm(-1)^{\frac{m'-1}2\frac{n'-1}2},

where the \pm sign is equal to + if m\ge0 or n\ge0 and is equal to - if m<0 and n<0.

There is also equivalent non-symmetric version of quadratic reciprocity that holds for every pair of integers m,n (not necessarily relatively prime):

\left(\frac mn\right)=(-1)^{\frac{m'-1}2\frac{n'-1}2}\left(\frac{n}{|m|}\right).

For any integer n let n^*=(-1)^{(n'-1)/2}n. Then we have another equivalent non-symmetric version that states

\left(\frac{m^*}{n}\right)=\left(\frac{n}{|m|}\right)

for every pair of integers m,n.

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 n we have

\left(\frac{-1}{n}\right)=(-1)^{\frac{n'-1}{2}}

and for any odd integer n it's

\left(\frac{2}{n}\right)=(-1)^{\frac{n^2-1}{8}}.

Connection to Dirichlet characters[edit]

If a\not\equiv3\pmod 4 and a\ne0, the map \chi(n)=\left(\tfrac an\right) is a real Dirichlet character of modulus \begin{cases}4|a|,&a\equiv2\pmod 4,\\|a|,&\text{otherwise.}\end{cases} Conversely, every real Dirichlet character can be written in this form with a\equiv0,1\pmod 4 (for a\equiv2\pmod 4 it's \left(\tfrac{a}{n}\right)=\left(\tfrac{4a}{n}\right)).

In particular, primitive real Dirichlet characters \chi are in a 1–1 correspondence with quadratic fields F=\mathbb Q(\sqrt m), where m is a nonzero square-free integer (we can include the case \mathbb Q(\sqrt1)=\mathbb Q to represent the principal character, even though it is not a proper quadratic field). The character \chi can be recovered from the field as the Artin symbol \left(\tfrac{F/\mathbb Q}\cdot\right): that is, for a positive prime p, the value of \chi(p) depends on the behaviour of the ideal (p) in the ring of integers O_F:

\chi(p)=\begin{cases}0,&(p)\text{ is ramified,}\\1,&(p)\text{ splits,}\\-1,&(p)\text{ is inert.}\end{cases}

Then \chi(n) equals the Kronecker symbol \left(\tfrac Dn\right), where

D=\begin{cases}m,&m\equiv1\pmod 4,\\4m,&m\equiv2,3\pmod 4\end{cases}

is the discriminant of F. The conductor of \chi is |D|.

Similarly, if n>0, the map \chi(a)=\left(\tfrac an\right) is a real Dirichlet character of modulus \begin{cases}4n,&n\equiv2\pmod 4,\\n,&\text{otherwise.}\end{cases} However, not all real characters can be represented in this way, for example the character \left(\tfrac{-4}\cdot\right) cannot be written as \left(\tfrac\cdot n\right) for any n. By the law of quadratic reciprocity, we have \left(\tfrac\cdot n\right)=\left(\tfrac{n^*}\cdot\right). A character \left(\tfrac a\cdot\right) can be represented as \left(\tfrac\cdot n\right) if and only if its odd part a'\equiv1\pmod4, in which case we can take n=|a|.

References[edit]

This article incorporates material from Kronecker symbol on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.