Legendre Symbol and Quadratic Reciprocity
Let p and q be odd primes. Using the Legendre symbol, the quadratic reciprocity law can be stated concisely:
Many proofs of quadratic reciprocity are based on Legendre's formula
In addition, several alternative expressions for the Legendre symbol were devised in order to produce various proofs of the quadratic reciprocity law.
- Gauss introduced the quadratic Gauss sum and used the formula
- in his fourth and sixth proofs of quadratic reciprocity.
- Kronecker's proof first establishes that
- Reversing the roles of p and q, he obtains the relation between and
- One of Eisenstein's proofs begins by showing that
- Using certain elliptic functions instead of the sine function, Eisenstein was able to prove cubic and quartic reciprocity as well.
Read more about this topic: Legendre Symbol
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