In number theory, the Kronecker symbol, written as ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} or ( a | n ) {\displaystyle (a|n)} , is a generalization of the Jacobi symbol to all integers n {\displaystyle n} . It was introduced by Leopold Kronecker (1885, page 770).
Definition Let n {\displaystyle n} be a non-zero integer, with prime factorization
n = u ⋅ p 1 e 1 ⋯ p k e k , {\displaystyle n=u\cdot p_{1}^{e_{1}}\cdots p_{k}^{e_{k}},}
where u {\displaystyle u} is a unit (i.e., u = ± 1 {\displaystyle u=\pm 1} ), and the p i {\displaystyle p_{i}} are primes. Let a {\displaystyle a} be an integer. The Kronecker symbol ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} is defined by
( a n ) := ( a u ) ∏ i = 1 k ( a p i ) e i . {\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 p i {\displaystyle p_{i}} , the number ( a p i ) {\displaystyle \left({\frac {a}{p_{i}}}\right)} is simply the usual Legendre symbol. This leaves the case when p i = 2 {\displaystyle p_{i}=2} . We define ( a 2 ) {\displaystyle \left({\frac {a}{2}}\right)} by
( a 2 ) := { 0 if a is even, 1 if a ≡ ± 1 ( mod 8 ) , − 1 if a ≡ ± 3 ( mod 8 ) . {\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 ( a u ) {\displaystyle \left({\frac {a}{u}}\right)} is simply 1 {\displaystyle 1} when u = 1 {\displaystyle u=1} . When u = − 1 {\displaystyle u=-1} , we define it by
… excerpt ends here. Continue reading the full article.
