system mod In other words, there are numbers in the set and this set constitutes a representative set for The numbers , used in the original version of the lemma, are a 1/2 system (mod ). Constructing a system is straightforward: let be a representative set for Pick any and remove the numbers congruent to from . Pick from and remove the numbers congruent to Repeat until is exhausted. Then is a system mod The lemma for nth powers Gauss's lemma may be extended to the th power residue symbol as follows. Let be a primitive th root of unity, a prime ideal, (i.e. is coprime to both and ) and let be a system mod Then for each , , there are integers , unique (mod ), and , unique (mod ), such that and the th-power residue symbol is given by the formula The classical lemma for the quadratic Legendre symbol is the special case , , , if , if . Proof The proof of the th-power lemma uses the same ideas that were used in the proof of the quadratic lemma. The existence of the integers and , and their uniqueness (mod ) and (mod ), respectively, come from the fact that is a representative set. Assume that = = , i.e. and Then Because and are coprime both sides can be divided by , giving which, since is a system, implies and , showing that is a permutation of the set . Then on the one hand, by the definition of the power residue symbol, and on the other hand, since is a permutation, so and since for all , and are coprime, can be cancelled from both sides of the congruence, and the theorem follows from the fact that no two distinct th roots of unity can be congruent (mod ). Relation to the transfer in group theory Let be the multiplicative group of nonzero residue classes in , and let be the subgroup {+1, −1}. Consider the following coset representatives of in , Applying the machinery of the transfer to this collection of coset representatives, we obtain the transfer homomorphism which turns out to be the map that sends to , where and are as in the statement of the lemma. Gauss's lemma may then be viewed as a computation that explicitly identifies this homomorphism as being the quadratic residue character. See also