Page "Quadratic residue" Paragraph 49

This phenomenon can best be described using the vocabulary of abstract algebra.

The congruence classes relatively prime to the modulus are a group under multiplication, called the group of units of the ring Z / nZ, and the squares are a subgroup of it.

Different nonresidues may belong to different cosets, and there is no simple rule that predicts which one their product will be in.

Modulo a prime, there is only the subgroup of squares and a single coset.