Here is a proof of quadratic reciprocity, as short and self-contained as I managed to do.
I use the ring Z/pZ[X]/(1+X+...+X^(p-1)) instead of the cyclotomic ring, in order to get directly uniqueness of the representative of an element of the ring as a polynomial of degree at most p-2, which is used at the end in order to ensure that the zero element of Z/pZ[X]/(1+X+...+X^(p-1)) cannot be represented by a non-zero element of Z/qZ.
In the cyclotomic ring Z[zeta] for a primitive p-th root zeta of the unity, the uniqueness property needs the irreducibility of 1 + X + ... X^(p-1) over the rationals, which is quite simple to prove but needs some extra work.
The proof uses Gauss sums, which is classical. The sign of these sums is not needed here, only the value of their square is used in our computations.
24 сен 2024
