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.

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 is 1 or −1), and the pi 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 pi, the number (a|pi) is simply the usual Legendre symbol. This leaves the case when pi = 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 n.

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 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[edit]

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[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.

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.