查看“︁雅可比符号”︁的源代码

在数论中，'''雅可比符号'''是[[勒让德符号]]的一种推广，首先由[[普鲁士]][[数学家]][[卡尔·雅可比]]在1837年引进<ref>C.G.J.Jacobi "Uber die Kreisteilung und ihre Anwendung auf die Zahlentheorie", ''Bericht Ak. Wiss. Berlin'' (1837) pp 127-136</ref>。雅可比符号在数论中的各个分支中都有应用，尤其是在计算数论的[[素性检验]]、[[大数分解]]以及[[密码学]]中有重要作用。 

==定义==

[[勒让德符号]]<math>(\tfrac{a}{p})</math>是对于所有的正[[整数]] <math>a</math> 和所有的[[素数]] <math>p</math> 定义的。
:{| align=left border="0"
|rowspan=3| <math>\left(\frac{a}{p}\right) =\begin{cases}\;\;\,0 \\+1 \\-1 \end{cases}</math>
| 如果p整除a；
|-
| 如果存在整数 <math>X</math> 使得 <math>X^2 \equiv a \pmod{p}</math> 且p不整除a
|-
| 如果不存在整数 <math>X</math> 使得 <math>X^2 \equiv a \pmod{p}</math> 
|}
:.
:；
:.
:.
当<math>(\frac{a}{p})= 1</math> 时，稱<math>a</math> 是模<math>p</math>的二次剩餘；当<math>(\frac{a}{p})=- 1</math> 时，稱<math>a</math> 是模<math>p</math>的二次非剩餘。

运用勒让德符号计算时要将 <math>a</math> 分解成标准形式，计算上十分麻烦，因此产生了'''雅可比符号'''：

设 <math>m</math> 是一个正[[奇数]]，其质因数分解式为 <math>m=\prod_{i=1}^s p_i</math>，并且正整数 <math>a</math> 满足 <math>(m,a) = 1</math> 那么定义<math>(\frac{a}{m})= \prod_{i=1}^s (\frac{a}{p_i})</math>。

==参见==

*[[克罗内克符号]]，将雅可比符号推广到任意自然数上。

==注释==

{{reflist}}

==参考来源==

*{{citation
  | last1 = Bach  | first1 = Eric
  | last2 = Shallit | first2 = Jeffrey 
  | title = Algorithmic Number Theory (Vol I: Efficient Algorithms)
  | publisher = [[The MIT Press]]
  | location = Cambridge
  | date = 1996
  | isbn = 9780262024051}}

*{{citation
  | last1 = Lemmermeyer  | first1 = Franz
  | title = Reciprocity Laws: from Euler to Eisenstein
  | publisher = [[Springer]]
  | location = Berlin
  | date = 2000
  | isbn = 9783540669579}}

*{{citation
  | last1 = Ireland  | first1 = Kenneth
  | last2 = Rosen  | first2 = Michael
  | title = A Classical Introduction to Modern Number Theory (Second edition)
  | publisher = [[Springer]]
  | location = New York
  | date = 1990
  | isbn = 0-387-97329-X}}

*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Maser | first2 = H. (translator into German)  
  | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | date = 1965
  | isbn = 0-8284-0191-8}}

*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Clarke | first2 = Arthur A. (translator into English)  
  | title = Disquisitiones Arithemeticae (Second, corrected edition)
  | publisher = [[Springer]]
  | location = New York
  | date = 1986
  | isbn = 0387962549}}

==外部链接==
* [https://web.archive.org/web/20080720165638/http://www.math.fau.edu/Richman/jacobi.htm Calculate Jacobi symbol] gives a display like the ones in the examples.

[[Category:同余]]
[[Category:数学符号]]