Quadratic reciprocity
The law of quadratic reciprocity, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows to determine the solvability of any quadratic equation in modular arithmetic.Suppose p and q are two different odd primes. If at least one of them is congruent to 1 modulo 4, then the congruence
- <math>x^2\equiv p\ ({\rm mod}\ q)</math>
- <math>y^2\equiv q\ ({\rm mod}\ p)</math>
- <math>x^2\equiv p\ ({\rm mod}\ q)</math>
- <math>y^2\equiv q\ ({\rm mod}\ p)</math>
Using the Legendre symbol (p/q), these statements may be summarized as
- <math>(p/q)\cdot(q/p)=(-1)^{(p-1)/2\cdot(q-1)/2}.</math>
For example taking p to be 11 and q to be 19, we can relate (11/19) to (19/11) which is (8/11). To proceed further we may need to know the supplementary laws computing (2/q) and (-1/q) explicitly. For example
- <math>(-1/q)= (-1)^{(q-1)/2}.</math>
Using this we relate (8/11) to (-3/11) to (3/11) to (11/3) to (2/3) to (-1/3); and can complete the initial calculation.
In a book about reciprocity laws published in 2000, Lemmermeyer collects literature citations for 196 different published proofs for the quadratic reciprocity law.
There are cubic, quartic (biquadratic) and other higher reciprocity laws[?]; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws).
External links
- A "play" comparing two proofs of the quadratic reciprocity law (http://www.math.nmsu.edu/~history/schauspiel/schauspiel.html)
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WALT WHITMAN
O CAPTAIN! MY CAPTAIN!
O Captain! My Captain! our fearful trip is done
The port is near, the bells I hear, the people all exulting,
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Where on the deck my Captain lies,
Rise up--for you the flag is flung--for you the bugle trills--
For you they call, the swaying mass, their eager faces turning.
This arm beneath your head!
You've fallen cold and dead.
My Captain does not answer, his lips are pale and still;
The ship is anchor'd safe and sound, its voyage closed and done;
Exult, O shores! and ring, O bells!
Walk the deck my Captain lies,
NOTES
ANNE DUDLEY BRADSTREET
"One wishes she were more winning: yet there is no gainsaying that she
observer; often dexterous in her verse--catching betimes upon epithets
always considered herself an exile. In 1644 her husband moved deeper
England" wrote her poems and brought up a family of eight children.
in America."
7. delectable giving pleasure.
MICHAEL WIGGLESWORTH (1631-1705)
"He was, himself, in nearly all respects, the embodiment of what was
of all offences, of all defects, there are in his poetry an irresistible
the prose of John Bunyan."
M. C. TYLER.
Born in England, he was brought to America at the age of seven. He
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