Quadratic reciprocity
| If p,q are odd primes, then |
( |
p
q |
)( |
q
p |
) |
= (-1)(p-1)/2(q-1)/2,
where |
Another way of phrasing this, which is perhaps more useful, is that
| ( |
p
q |
) |
= |
( |
q
p |
) |
unless both p and q are primes
of the form 4k+3 |
Calculating the value of a Legendre symbol using Quadratic Reciprocity
Example, using Quadratic Reciprocity,
| ( |
3
983 |
) |
= (-1) |
( |
983
3 |
) |
because both 3 and 983 are
primes of the form 4k+3 |
| = (-1) |
( |
2
3 |
) |
= (-1)(-1) = 1 |
| More generally, Quadratic
Reciprocity can be used to find |
( |
3
p |
) |
as follows: |
| ( |
3
p |
) |
= |
( |
p
3 |
) |
if p≡1 (mod 4), and
|
( |
3
p |
) |
= (-1) |
( |
p
3 |
) |
if p≡-1 (mod 4). |
Since 1 is the only quadratic residue (mod 3), and -1 is the only quadratic
non-residue (mod 3), it follows that
| ( |
3
p |
) |
= |
{ |
1 if p≡±1 (mod 12),
-1 if p≡�5 (mod 12), |
Internet references
Wikipedia: Law
of quadratic reciprocity
Numericana, Final Answers:
Quadratic Reciprocity
Related pages in this website
|
Jacobi symbol — |
( |
a
n |
) |
, where a is any integer, and n is a
positive integer greater than 2, an extension of the Legendre symbol. |
|
Kronecker symbol — |
( |
a
n |
) |
, where a and n are any integers, an
extension of the Jacobi symbol. |
Fermat's Little Theorem
Euler's Totient Theorem
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