used in many, but by no means all, of the known proofs of quadratic reciprocity. For example, Gotthold Eisenstein used Gauss's lemma to prove that if is an odd prime then and used this formula to prove quadratic reciprocity. By using elliptic rather than circular functions, he proved the cubic and quartic reciprocity laws. Leopold Kronecker used the lemma to show that Switching and immediately gives quadratic reciprocity. It is also used in what are probably the simplest proofs of the "second supplementary law" Higher powers Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of in , the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity. nth power residue symbol Let k be an algebraic number field with ring of integers and let be a prime ideal. The ideal norm of is defined as the cardinality of the residue class ring. Since is prime this is a finite field , so the ideal norm is . Assume that a primitive th root of unity and that and are coprime (i.e. ). Then no two distinct th roots of unity can be congruent modulo . This can be proved by contradiction, beginning by assuming that mod , . Let such that mod , and . From the definition of roots of unity, and dividing by gives Letting and taking residues mod , Since and are coprime, mod but under the assumption, one of the factors on the right must be zero. Therefore, the assumption that two distinct roots are congruent is false. Thus the residue classes of containing the powers of are a subgroup of order of its (multiplicative) group of units, Therefore, the order of is a multiple of , and There is an analogue of Fermat's theorem in . If for , then and since mod , is well-defined and congruent to a unique th root of unity ζns. This root of unity is called the th-power residue symbol for and is denoted by It can be proven that if and only if there is an such that mod . 1/n systems Let be the multiplicative group of the th roots of unity, and let be representatives of the cosets of Then is called a