Kronecker symbol

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

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 is 1 or −1), 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} = \begin{cases} 0 &\mbox{ if } a\mbox{ is even,} \\ \displaystyle \left(\frac 2a\right) &\mbox{ if } a \mbox{ is odd.} \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}

and

\left(\frac a1\right)=1.

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

Some authors only define the Kronecker symbol for more restricted values; for example, a congruent to 0 or 1 mod 4 and n positive.

Properties

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 $n=-1$ and one of $a,b$ is zero.
$\left(\tfrac a{nm}\right)=\left(\tfrac an\right)\left(\tfrac am\right)$ unless $a=-1$ and one of $n,m$ is zero.
For $n>0$ , we have $\left(\tfrac an\right)=\left(\tfrac bn\right)$ whenever $a\equiv b\mod\begin{cases}4n,&n\equiv2\pmod 4,\\n&\text{otherwise.}\end{cases}$ If additionally $a,b$ have the same sign, the same also holds for $n<0$ .
For $a\not\equiv3\pmod4$ , $a\ne0$ , we have $\left(\tfrac an\right)=\left(\tfrac am\right)$ whenever $n\equiv m\mod\begin{cases}4|a|,&a\equiv2\pmod 4,\\|a|&\text{otherwise.}\end{cases}$

Quadratic reciprocity

The Kronecker symbol also satisfies the following version of quadratic reciprocity.

For any nonzero integer $n$ , let $n'$ denote its odd part: $n=2^en'$ where $n'$ is odd (for $n=0$ , we put $0'=1$ ). Let $n^*=(-1)^{(n'-1)/2}n$ . Then if $n\ge0$ or $m\ge0$ , then

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

Connection to Dirichlet characters

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.

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

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.

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