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 \mathcal{O}_k, and let \mathfrak{p} \subset \mathcal{O}_k be a
prime ideal. The
ideal norm \mathrm{N} \mathfrak{p} of \mathfrak{p} is defined as the cardinality of the residue class ring. Since \mathfrak{p} is prime this is a
finite field \mathcal{O}_k / \mathfrak{p}, so the ideal norm is \mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}|. Assume that a primitive th
root of unity \zeta_n\in\mathcal{O}_k, and that and \mathfrak{p} are
coprime (i.e. n\not\in \mathfrak{p}). Then no two distinct th roots of unity can be congruent modulo \mathfrak{p}. This can be proved by contradiction, beginning by assuming that \zeta_n^r\equiv\zeta_n^s mod \mathfrak{p}, . Let such that \zeta_n^t\equiv 1 mod \mathfrak{p}, and . From the definition of roots of unity, :x^n-1=(x-1)(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}), and dividing by gives :x^{n-1}+x^{n-2}+\dots +x + 1 =(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}). Letting and taking residues mod \mathfrak{p}, :n\equiv(1-\zeta_n)(1-\zeta_n^2)\dots(1-\zeta_n^{n-1})\pmod{\mathfrak{p}}. Since and \mathfrak{p} are coprime, n\not\equiv 0 mod \mathfrak{p}, 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 \mathcal{O}_k / \mathfrak{p} containing the powers of are a subgroup of order of its (multiplicative) group of units, (\mathcal{O}_k/\mathfrak{p}) ^\times = \mathcal{O}_k /\mathfrak{p}- \{0\}. Therefore, the order of (\mathcal{O}_k/\mathfrak{p})^ \times is a multiple of , and : \begin{align} \mathrm{N} \mathfrak{p} &= |\mathcal{O}_k / \mathfrak{p}| \\ &= \left |(\mathcal{O}_k / \mathfrak{p} )^\times \right| + 1 \\ &\equiv 1 \pmod{n}. \end{align} There is an analogue of Fermat's theorem in \mathcal{O}_k. If \alpha \in \mathcal{O}_k for \alpha\not\in \mathfrak{p}, then :\alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \pmod{\mathfrak{p} }, and since \mathrm{N} \mathfrak{p} \equiv 1 mod , :\alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\pmod{\mathfrak{p} } is well-defined and congruent to a unique th root of unity ζ
ns. This root of unity is called the
th-power residue symbol for \mathcal{O}_k, and is denoted by : \begin{align} \left(\frac{\alpha}{\mathfrak{p} }\right)_n &= \zeta_n^s \\ &\equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}. \end{align} It can be proven that :\left(\frac{\alpha}{\mathfrak{p} }\right)_n= 1 if and only if there is an \eta \in\mathcal{O}_k such that mod \mathfrak{p}.
1/n systems Let \mu_n = \{1,\zeta_n,\zeta_n^2,\dots,\zeta_n^{n-1}\} be the multiplicative group of the th roots of unity, and let A=\{a_1, a_2,\dots,a_m\} be representatives of the cosets of (\mathcal{O}_k / \mathfrak{p})^\times/\mu_n. Then is called a
system mod \mathfrak{p}. In other words, there are mn=\mathrm{N} \mathfrak{p} -1 numbers in the set A\mu=\{ a_i \zeta_n^j\;:\; 1 \le i \le m, \;\;\;0 \le j \le n-1\}, and this set constitutes a representative set for (\mathcal{O}_k / \mathfrak{p})^\times. 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 (\mathcal{O}_k / \mathfrak{p})^\times. Pick any a_1\in M and remove the numbers congruent to a_1, a_1\zeta_n, a_1\zeta_n^2, \dots, a_1\zeta_n^{n-1} from . Pick from and remove the numbers congruent to a_2, a_2\zeta_n, a_2\zeta_n^2, \dots, a_2\zeta_n^{n-1} Repeat until is exhausted. Then is a system mod \mathfrak{p}.
The lemma for nth powers Gauss's lemma may be extended to the th
power residue symbol as follows. Let \zeta_n\in \mathcal{O}_k be a primitive th root of unity, \mathfrak{p} \subset \mathcal{O}_k a prime ideal, \gamma \in \mathcal{O}_k, \;\;n\gamma\not\in\mathfrak{p}, (i.e. \mathfrak{p} is coprime to both and ) and let be a system mod \mathfrak{p}. Then for each , , there are integers , unique (mod ), and , unique (mod ), such that :\gamma a_i \equiv \zeta_n^{b(i)}a_{\pi(i)} \pmod{\mathfrak{p}}, and the th-power residue symbol is given by the formula :\left(\frac{\gamma}{\mathfrak{p} }\right)_n = \zeta_n^{b(1)+b(2)+\dots+b(m)}. 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. :\gamma a_i \equiv \zeta_n^r a_p \pmod{\mathfrak{p}} and :\gamma a_j \equiv \zeta_n^s a_p \pmod{\mathfrak{p}}. Then :\zeta_n^{s-r}\gamma a_i \equiv \zeta_n^s a_p \equiv \gamma a_j\pmod{\mathfrak{p}} Because and \mathfrak{p} are coprime both sides can be divided by , giving :\zeta_n^{s-r} a_i \equiv a_j\pmod{\mathfrak{p}}, 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, : \begin{align} (\gamma a_1)(\gamma a_2)\dots(\gamma a_m) &= \gamma^{\frac{\mathrm{N} \mathfrak{p} -1}{n}} a_1 a_2\dots a_m \\ &\equiv \left(\frac{\gamma}{\mathfrak{p} }\right)_n a_1 a_2\dots a_m \pmod{\mathfrak{p}}, \end{align} and on the other hand, since is a permutation, : \begin{align} (\gamma a_1)(\gamma a_2)\dots(\gamma a_m) &\equiv {\zeta_n^{b(1)}a_{\pi(1)}} {\zeta_n^{b(2)}a_{\pi(2)}}\dots{\zeta_n^{b(m)}a_{\pi(m)}} &\pmod{\mathfrak{p}}\\ &\equiv \zeta_n^{b(1)+b(2)+\dots+b(m)}a_{\pi(1)} a_{\pi(2)}\dots a_{\pi(m)} &\pmod{\mathfrak{p}}\\ &\equiv \zeta_n^{b(1)+b(2)+\dots+b(m)} a_1 a_2\dots a_m &\pmod{\mathfrak{p}}, \end{align} so : \left(\frac{\gamma}{\mathfrak{p} }\right)_n a_1 a_2\dots a_m \equiv \zeta_n^{b(1)+b(2)+\dots+b(m)} a_1 a_2\dots a_m \pmod{\mathfrak{p}}, and since for all , and \mathfrak{p} are coprime, can be cancelled from both sides of the congruence, :\left(\frac{\gamma}{\mathfrak{p} }\right)_n \equiv \zeta_n^{b(1)+b(2)+\dots+b(m)} \pmod{\mathfrak{p}}, and the theorem follows from the fact that no two distinct th roots of unity can be congruent (mod \mathfrak{p}). == Relation to the transfer in group theory ==