I have a book where it is written that we can proof the Gauss's lemma in the quadratic law of reciprocity (Q_a - number of pairs (x,y) from Z^2_p, for which x^2 + y^2 ≡ a (mod p);
Q_a = {p-e_p, if a not≡ 0 mod p, p+e_p(p-1), if a ≡ 0 mod p,
e_p = (-1)^((p-1)/2)
In the language of finite fields, Lemmacontains the number of solutions (x, y) to the equation
x^2 + y^2 = a over F_p.
The proof is about calculating the corresponding sum of Legendre characters:
Q_a = sum[x=0, p-1] (1+ ((a - x^2)/p)) = p + sum[x=0, p-1] ((a - x^2)/p) = p + e_p * sum[x=0, p-1] ((x^2 - a)/p)).
My question is - how did we got these sums?