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Quadratic residue
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===Composite modulus not a prime power=== The basic fact in this case is :if ''a'' is a residue modulo ''n'', then ''a'' is a residue modulo ''p''<sup>''k''</sup> for ''every'' prime power dividing ''n''. :if ''a'' is a nonresidue modulo ''n'', then ''a'' is a nonresidue modulo ''p''<sup>''k''</sup> for ''at least one'' prime power dividing ''n''. Modulo a composite number, the product of two residues is a residue. The product of a residue and a nonresidue may be a residue, a nonresidue, or zero. <blockquote> For example, from the table for modulus 6 '''1''', 2, '''3''', '''4''', 5 (residues in '''bold'''). The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2. </blockquote> Also, the product of two nonresidues may be either a residue, a nonresidue, or zero. <blockquote> For example, from the table for modulus 15 '''1''', 2, 3, '''4''', 5, '''6''', 7, 8, '''9''', '''10''', 11, 12, 13, 14 (residues in '''bold'''). The product of the nonresidues 2 and 8 is the residue 1, whereas the product of the nonresidues 2 and 7 is the nonresidue 14. </blockquote> This phenomenon can best be described using the vocabulary of abstract algebra. The congruence classes relatively prime to the modulus are a [[Group (mathematics)|group]] under multiplication, called the [[group of units]] of the [[Ring (mathematics)|ring]] <math>(\mathbb{Z}/n\mathbb{Z})</math>, and the squares are a [[subgroup]] of it. Different nonresidues may belong to different [[coset]]s, 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. The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring <math>(\mathbb{Z}/n\mathbb{Z})</math>, which has [[zero divisor]]s for composite ''n''. For this reason some authors<ref>e.g., {{harvnb|Ireland|Rosen|1990|p=50}}</ref> add to the definition that a quadratic residue ''a'' must not only be a square but must also be [[relatively prime]] to the modulus ''n''. (''a'' is coprime to ''n'' if and only if ''a''<sup>2</sup> is coprime to ''n''.) Although it makes things tidier, this article does not insist that residues must be coprime to the modulus.
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