Quadratic Reciprocity Ii: the Proofs

Let p be a prime number; for x, y ∈ Rn, we will write x ≡ y (mod p) to mean that there exists a z ∈ Rn such that x− y = pz. Otherwise put, this is congruence modulo the principal ideal pRn of Rn. Since Z ⊂ Rn, if x and y are ordinary integers, the notation x ≡ y (mod p) is ambiguous: interpreting it as a usual congruence in the integers, it means that there exists an integer n such that x− y = pn; and interpreting it as a congruence in Rn, it means that x− y = pz for some z ∈ Rn. The key technical point is that these two notions of congruence are in fact the same: