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quadratic reciprocity
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Definition/Summary
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| A number n is a quadratic residue mod m if there exists some number a which, squared mod m, gives n. |
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Equations
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Definition of the Legendre symbol, for any number a and for any odd prime p:
[tex]\left(\frac ap\right)=\begin{cases}
0&p|a\\
1&\exists n:n^2\equiv a\pmod p\\
-1&\nexists n:n^2\equiv a\pmod p
\end{cases}[/tex]
The Legendre symbol is multiplicative:
[tex]\left(\frac{ab}{p}\right)=\left(\frac ap\right)\left(\frac bp\right)[/tex]
The Law of Quadratic Reciprocity, for any odd primes p and q:
[tex]\left(\frac qp\right)=(-1)^{(p-1)(q-1)/4}\left(\frac pq\right)[/tex] |
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Scientists
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| Euler, Gauss, Legendre, Jacobi, Hilbert |
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Recent forum threads on quadratic reciprocity
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Breakdown
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Mathematics
> Number Theory
>> Reciprocity Theorems
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Extended explanation
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For example, 0 1 4 5 6 and 9 are quadratic residues mod 10 because the squares of "ordinary" numbers (which are "base 10") can end in 0 1 4 5 6 or 9.
2 3 7 and 8 are not quadratic residues mod 10.
The law of Quadratic Reciprocity, of course, does not apply mod 10, because 10 is not a prime.
A generalisation of the Legendre symbol for odd non-primes p is the Jacobi symbol.
There is also a Hilbert symbol.
There are extensions of the law of Quadratic Reciprocity for non-prime p and q. |
Commentary
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