User:Virginia-American/Sandbox/Power residue symbol

In algebraic number theory the n-th power residue symbol (for an integer n > 2) is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws.

Background and notation
Let k be an algebraic number field with ring of integers   $$\mathcal{O}_k$$  that contains a primitive nth root of unity    $$\zeta_n\in\mathcal{O}_k.$$

Let   $$\mathfrak{p} \subset \mathcal{O}_k $$    be a prime ideal and assume that n and $$\mathfrak{p}$$ are coprime (i.e. $$n \not \in \mathfrak{p}$$.)

The norm of $$\mathfrak{p} $$  is defined as the cardinality of the residue class ring    $$ \mathcal{O}_k / \mathfrak{p}\;:\;\;\; \mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}|.$$   (since $$\mathfrak{p} $$  is prime this is a finite field)

There is an analogue of Fermat's theorem in  $$\mathcal{O}_k:$$  If    $$\alpha \in \mathcal{O}_k,\;\;\; \alpha\not\in \mathfrak{p},$$   then
 * $$\alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \pmod{\mathfrak{p} }.

$$

And finally,   $$\mathrm{N} \mathfrak{p} \equiv 1 \pmod{n}.$$ These facts imply that
 * $$\alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\pmod{\mathfrak{p} }

$$  is well-defined and congruent to a unique n-th root of unity &zeta;ns.

Definition
This root of unity is called the n-th power residue symbol for    $$\mathcal{O}_k,$$    and is denoted by



\left(\frac{\alpha}{\mathfrak{p} }\right)_n= \zeta_n^s \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}. $$

Properties
The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol:

\left(\frac{\alpha}{\mathfrak{p} }\right)_n = \begin{cases} 0 \mbox{ if } \alpha\in\mathfrak{p}\\ 1 \mbox{ if }\alpha\not\in\mathfrak{p}\mbox{ there is an } \eta \in\mathcal{O}_k\;\;\mbox{ such that } \;\;\alpha\equiv\eta^n\pmod{\mathfrak{p}}\\ \zeta \mbox{ where }\zeta^n=1\mbox{ and }\zeta \neq 1\mbox{ if there is no such }\eta \end{cases} $$

In all cases (zero and nonzero)



\left(\frac{\alpha}{\mathfrak{p} }\right)_n \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}. $$



\left(\frac{\alpha}{\mathfrak{p} }\right)_n \left(\frac{\beta}{\mathfrak{p} }\right)_n = \left(\frac{\alpha\beta}{\mathfrak{p} }\right)_n $$



\mbox{if }\alpha \equiv\beta\pmod{\mathfrak{p}} \mbox{ then } \left(\frac{\alpha}{\mathfrak{p} }\right)_n = \left(\frac{\beta}{\mathfrak{p} }\right)_n $$

Generalizations
The n-th power symbol may be extended to take non-prime ideals or non-zero elements as its "denominator", in the same way that the Jacobi symbol extends the Legendre symbol.

Any ideal $$\mathfrak{a}\subset\mathcal{O}_k$$ is the product of prime ideals, and in one way only:
 * $$\mathfrak{a} = \mathfrak{p}_1 \mathfrak{p}_2 \dots\mathfrak{p}_g.

$$

The n-th power symbol is extended multiplicatively:



\bigg(\frac{\alpha}{\mathfrak{a} }\bigg)_n = \left(\frac{\alpha}{\mathfrak{p}_1 }\right)_n \left(\frac{\alpha}{\mathfrak{p}_2 }\right)_n \dots \left(\frac{\alpha}{\mathfrak{p}_g }\right)_n. $$

If $$\beta\in\mathcal{O}_k$$ is not zero the symbol $$\left(\frac{\alpha}{\beta}\right)_n$$ is defined as

\left(\frac{\alpha}{\beta}\right)_n = \left(\frac{\alpha}{(\beta) }\right)_n, $$ where $$(\beta)$$ is the prinicpal ideal generated by $$\beta.$$

The properties of this symbol are analogous to those of the quadratic Jacobi symbol:

\mbox{If }\alpha\equiv\beta\pmod{\mathfrak{a}} \mbox{ then } \bigg(\frac{\alpha}{\mathfrak{a} }\bigg)_n = \left(\frac{\beta}{\mathfrak{a} }\right)_n. $$



\bigg(\frac{\alpha}{\mathfrak{a} }\bigg)_n \left(\frac{\beta}{\mathfrak{a} }\right)_n = \left(\frac{\alpha\beta}{\mathfrak{a} }\right)_n. $$



\left(\frac{\alpha}{\mathfrak{a} }\right)_n \left(\frac{\alpha}{\mathfrak{b} }\right)_n = \left(\frac{\alpha}{\mathfrak{ab} }\right)_n. $$



\mbox{If } \alpha\equiv\eta^n\pmod{\mathfrak{a}}\mbox{ then } \left(\frac{\alpha}{\mathfrak{a} }\right)_n =1. $$



\mbox{If } \left(\frac{\alpha}{\mathfrak{a} }\right)_n \neq1 \mbox{ then }\alpha \mbox{ is not an }n\mbox{-th power}\pmod{\mathfrak{a}}. $$



\mbox{If } \left(\frac{\alpha}{\mathfrak{a} }\right)_n =1 \mbox{ then }\alpha \mbox{ may or may not be an }n\mbox{-th power}\pmod{\mathfrak{a}}. $$