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 n^th 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 ζ_n^s.

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:

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

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

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

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

${\mbox{If }}\left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}\neq 1{\mbox{ then }}\alpha {\mbox{ is not an }}n{\mbox{-th power}}{\pmod {\mathfrak {a}}}.$

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

See also

• Artin symbol
• Hilbert symbol
• Gauss's lemma

References

• Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X

• Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66967-4 {{citation}}: Check |isbn= value: checksum (help)