Kronecker symbol

Let ${\displaystyle n}$ be a non-zero integer, with prime factorization

${\displaystyle n=u\cdot p_{1}^{e_{1}}\cdots p_{k}^{e_{k}},}$

where ${\displaystyle u}$ is a unit (i.e., ${\displaystyle u=\pm 1}$), and the ${\displaystyle p_{i}}$ are primes. Let ${\displaystyle a}$ be an integer. The Kronecker symbol ${\displaystyle \left({\frac {a}{n}}\right)}$ is defined by

${\displaystyle \left({\frac {a}{n}}\right)=\left({\frac {a}{u}}\right)\prod _{i=1}^{k}\left({\frac {a}{p_{i}}}\right)^{e_{i}}.}$

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

${\displaystyle \left({\frac {a}{2}}\right)={\begin{cases}0&{\mbox{if }}a{\mbox{ is even,}}\\1&{\mbox{if }}a\equiv \pm 1{\pmod {8}},\\-1&{\mbox{if }}a\equiv \pm 3{\pmod {8}}.\end{cases}}}$

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

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

Finally, we put

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

These extensions suffice to define the Kronecker symbol for all integer values ${\displaystyle a,n}$.

Some authors only define the Kronecker symbol for more restricted values; for example, ${\displaystyle a}$ congruent to ${\displaystyle 0,1{\bmod {4}}}$ and ${\displaystyle n>0}$.

The following is a table of values of Kronecker symbol ${\displaystyle \left({\frac {k}{n}}\right)}$ with 1 ≤ n, k ≤ 30.

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

• ${\displaystyle \left({\tfrac {a}{n}}\right)=\pm 1}$ if ${\displaystyle \gcd(a,n)=1}$, otherwise ${\displaystyle \left({\tfrac {a}{n}}\right)=0}$.
• ${\displaystyle \left({\tfrac {ab}{n}}\right)=\left({\tfrac {a}{n}}\right)\left({\tfrac {b}{n}}\right)}$ unless ${\displaystyle n=-1}$, one of ${\displaystyle a,b}$ is zero and the other one is negative.
• ${\displaystyle \left({\tfrac {a}{mn}}\right)=\left({\tfrac {a}{m}}\right)\left({\tfrac {a}{n}}\right)}$ unless ${\displaystyle a=-1}$, one of ${\displaystyle m,n}$ is zero and the other one has odd part (definition below) congruent to ${\displaystyle 3{\bmod {4}}}$.
• For ${\displaystyle n>0}$, we have ${\displaystyle \left({\tfrac {a}{n}}\right)=\left({\tfrac {b}{n}}\right)}$ whenever ${\displaystyle a\equiv b{\bmod {\begin{cases}4n,&n\equiv 2{\pmod {4}},\\n&{\text{otherwise.}}\end{cases}}}}$ If additionally ${\displaystyle a,b}$ have the same sign, the same also holds for ${\displaystyle n<0}$.
• For ${\displaystyle a\not \equiv 3{\pmod {4}}}$, ${\displaystyle a\neq 0}$, we have ${\displaystyle \left({\tfrac {a}{m}}\right)=\left({\tfrac {a}{n}}\right)}$ whenever ${\displaystyle m\equiv n{\bmod {\begin{cases}4|a|,&a\equiv 2{\pmod {4}},\\|a|&{\text{otherwise.}}\end{cases}}}}$

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 ${\displaystyle \left({\tfrac {a}{n}}\right)}$ for even ${\displaystyle n}$ can take values independently on whether ${\displaystyle a}$ is a quadratic residue or nonresidue modulo ${\displaystyle n}$.

If ${\displaystyle a\not \equiv 3{\pmod {4}}}$ and ${\displaystyle a\neq 0}$, the map ${\displaystyle \chi (n)=\left({\tfrac {a}{n}}\right)}$ is a real Dirichlet character of modulus ${\displaystyle {\begin{cases}4|a|,&a\equiv 2{\pmod {4}},\\|a|,&{\text{otherwise.}}\end{cases}}}$ Conversely, every real Dirichlet character can be written in this form with ${\displaystyle a\equiv 0,1{\pmod {4}}}$ (for ${\displaystyle a\equiv 2{\pmod {4}}}$ it's ${\displaystyle \left({\tfrac {a}{n}}\right)=\left({\tfrac {4a}{n}}\right)}$).

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

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

Then ${\displaystyle \chi (n)}$ equals the Kronecker symbol ${\displaystyle \left({\tfrac {D}{n}}\right)}$, where

${\displaystyle D={\begin{cases}m,&m\equiv 1{\pmod {4}},\\4m,&m\equiv 2,3{\pmod {4}}\end{cases}}}$

is the discriminant of ${\displaystyle F}$. The conductor of ${\displaystyle \chi }$ is ${\displaystyle |D|}$.

Similarly, if ${\displaystyle n>0}$, the map ${\displaystyle \chi (a)=\left({\tfrac {a}{n}}\right)}$ is a real Dirichlet character of modulus ${\displaystyle {\begin{cases}4n,&n\equiv 2{\pmod {4}},\\n,&{\text{otherwise.}}\end{cases}}}$ However, not all real characters can be represented in this way, for example the character ${\displaystyle \left({\tfrac {-4}{\cdot }}\right)}$ cannot be written as ${\displaystyle \left({\tfrac {\cdot }{n}}\right)}$ for any ${\displaystyle n}$. By the law of quadratic reciprocity, we have ${\displaystyle \left({\tfrac {\cdot }{n}}\right)=\left({\tfrac {n^{*}}{\cdot }}\right)}$. A character ${\displaystyle \left({\tfrac {a}{\cdot }}\right)}$ can be represented as ${\displaystyle \left({\tfrac {\cdot }{n}}\right)}$ if and only if its odd part ${\displaystyle a'\equiv 1{\pmod {4}}}$, in which case we can take ${\displaystyle n=|a|}$.

• Hilbert symbol

• 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. Vol. 97. Cambridge University Press . ISBN 978-0-521-84903-6. Zbl 1142.11001.

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