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| {{about|the symbol in number theory||Kronecker delta}}
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| In [[number theory]], the '''Kronecker symbol''', written as <math>\left(\frac an\right)</math> or (''a''|''n''), is a generalization of the [[Jacobi symbol]] to all [[integer]]s ''n''. It was introduced by [[Leopold Kronecker]].
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| ==Definition==
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| Let ''n'' be a non-zero integer, with [[prime factorization]]
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| :<math>n=u \cdot p_1^{e_1} \cdots p_k^{e_k},</math> | |
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| where ''u'' is a [[unit (ring theory)|unit]] (i.e., ''u'' is 1 or −1), and the ''p<sub>i</sub>'' are [[prime]]s. Let ''a'' be an integer. The Kronecker symbol (''a''|''n'') is defined by
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| :<math> \left(\frac{a}{n}\right) = \left(\frac{a}{u}\right) \prod_{i=1}^k \left(\frac{a}{p_i}\right)^{e_i}. </math>
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| For [[odd number|odd]] ''p<sub>i</sub>'', the number (''a''|''p<sub>i</sub>'') is simply the usual [[Legendre symbol]]. This leaves the case when ''p<sub>i</sub>'' = 2. We define (''a''|2) by
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| :<math> \left(\frac{a}{2}\right) =
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| \begin{cases}
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| 0 & \mbox{if }a\mbox{ is even,} \\
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| 1 & \mbox{if } a \equiv \pm1 \pmod{8}, \\
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| -1 & \mbox{if } a \equiv \pm3 \pmod{8}.
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| \end{cases}</math>
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| Since it extends the Jacobi symbol, the quantity (''a''|''u'') is simply 1 when ''u'' = 1. When ''u'' = −1, we define it by
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| :<math> \left(\frac{a}{-1}\right) = \begin{cases} -1 & \mbox{if }a < 0, \\ 1 & \mbox{if } a \ge 0. \end{cases} </math>
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| Finally, we put
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| :<math>\left(\frac a0\right)=\begin{cases}1&\text{if }a=\pm1,\\0&\text{otherwise.}\end{cases}</math>
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| These extensions suffice to define the Kronecker symbol for all integer values ''n''.
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| ==Properties==
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| The Kronecker symbol shares many basic properties of the Jacobi symbol, under certain restrictions:
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| *<math>\left(\tfrac an\right)=\pm1</math> if <math>\gcd(a,n)=1</math>, otherwise <math>\left(\tfrac an\right)=0</math>.
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| *<math>\left(\tfrac{ab}n\right)=\left(\tfrac an\right)\left(\tfrac bn\right)</math> unless <math>n=-1</math> and one of <math>a,b</math> is zero.
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| *<math>\left(\tfrac a{nm}\right)=\left(\tfrac an\right)\left(\tfrac am\right)</math> unless <math>a=-1</math> and one of <math>n,m</math> is zero.
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| *For <math>n>0</math>, we have <math>\left(\tfrac an\right)=\left(\tfrac bn\right)</math> whenever <math>a\equiv b\mod\begin{cases}4n,&n\equiv2\pmod 4,\\n&\text{otherwise.}\end{cases}</math> If additionally <math>a,b</math> have the same sign, the same also holds for <math>n<0</math>.
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| *For <math>a\not\equiv3\pmod4</math>, <math>a\ne0</math>, we have <math>\left(\tfrac an\right)=\left(\tfrac am\right)</math> whenever <math>n\equiv m\mod\begin{cases}4|a|,&a\equiv2\pmod 4,\\|a|&\text{otherwise.}\end{cases}</math>
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| === Quadratic reciprocity ===
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| The Kronecker symbol also satisfies the following version of [[quadratic reciprocity]].
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| For any nonzero integer <math>n</math>, let <math>n'</math> denote its odd part: <math>n=2^en'</math> where <math>n'</math> is odd (for <math>n=0</math>, we put <math>0'=1</math>). Let <math>n^*=(-1)^{(n'-1)/2}n</math>. Then if <math>n\ge0</math> or <math>m\ge0</math>, then
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| : <math>\left(\frac nm\right)=\left(\frac{m^*}n\right)=(-1)^{\frac{n'-1}2\frac{m'-1}2}\left(\frac mn\right).</math>
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| ==Connection to Dirichlet characters==
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| If <math>a\not\equiv3\pmod 4</math> and <math>a\ne0</math>, the map <math>\chi(n)=\left(\tfrac an\right)</math> is a real [[Dirichlet character]] of modulus <math>\begin{cases}4|a|,&a\equiv2\pmod 4,\\|a|,&\text{otherwise.}\end{cases}</math> Conversely, every real Dirichlet character can be written in this form.
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| In particular, ''primitive'' real Dirichlet characters <math>\chi</math> are in a 1–1 correspondence with [[quadratic field]]s <math>F=\mathbb Q(\sqrt m)</math>, where ''m'' is a nonzero [[square-free integer]] (we can include the case <math>\mathbb Q(\sqrt1)=\mathbb Q</math> to represent the principal character, even though it is not a proper quadratic field). The character <math>\chi</math> can be recovered from the field as the [[Artin symbol]] <math>\left(\tfrac{F/\mathbb Q}\cdot\right)</math>: that is, for a positive prime ''p'', the value of <math>\chi(p)</math> depends on the behaviour of the ideal <math>(p)</math> in the [[ring of integers]] <math>O_F</math>:
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| :<math>\chi(p)=\begin{cases}0,&(p)\text{ is ramified,}\\1,&(p)\text{ splits,}\\-1,&(p)\text{ is inert.}\end{cases}</math> | |
| Then <math>\chi(n)</math> equals the Kronecker symbol <math>\left(\tfrac Dn\right)</math>, where
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| :<math>D=\begin{cases}m,&m\equiv1\pmod 4,\\4m,&m\equiv2,3\pmod 4\end{cases}</math> | |
| is the [[discriminant of an algebraic number field|discriminant]] of ''F''. The conductor of <math>\chi</math> is <math>|D|</math>.
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| Similarly, if <math>n>0</math>, the map <math>\chi(a)=\left(\tfrac an\right)</math> is a real Dirichlet character of modulus <math>\begin{cases}4n,&n\equiv2\pmod 4,\\n,&\text{otherwise.}\end{cases}</math> However, not all real characters can be represented in this way, for example the character <math>\left(\tfrac{-4}\cdot\right)</math> cannot be written as <math>\left(\tfrac\cdot n\right)</math> for any ''n''. By the law of quadratic reciprocity, we have <math>\left(\tfrac\cdot n\right)=\left(\tfrac{n^*}\cdot\right)</math>. A character <math>\left(\tfrac a\cdot\right)</math> can be represented as <math>\left(\tfrac\cdot n\right)</math> if and only if its odd part <math>a'\equiv1\pmod4</math>, in which case we can take <math>n=|a|</math>.
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| ==References==
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| * {{cite book | last1=Montgomery | first1=Hugh L | author1-link=Hugh Montgomery (mathematician) | last2=Vaughan | first2=Robert C. | author2-link=Bob Vaughan | title=Multiplicative number theory. I. Classical theory | series=Cambridge Studies in Advanced Mathematics | volume=97 | publisher=[[Cambridge University Press ]] | year=2007 | isbn=0-521-84903-9 | zbl=1142.11001 }}
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| {{PlanetMath attribution|id=6108|title=Kronecker symbol}}
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| [[Category:Modular arithmetic]]
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