Power residue symbol

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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^[1] reciprocity laws.^[2]

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:

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

In all cases (zero and nonzero)

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

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

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

Relation to the Hilbert symbol

The n-th power residue symbol is related to the Hilbert symbol $(\cdot ,\cdot )_{{{\mathfrak {p}}}}$ for the prime ${\mathfrak {p}}$ by

$\left({\frac {\alpha }{{\mathfrak {p}}}}\right)_{n}=\left({\pi ,\alpha }\right)_{{{\mathfrak {p}}}}$

in the case ${\mathfrak {p}}$ coprime to n, where $\pi$ is any uniformising element for the local field $K_{{{\mathfrak {p}}}}$ .^[3]

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 principal ideal generated by $\beta .$

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

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

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

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

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

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

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

Power reciprocity law

The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as^[4]

$\left({{\frac {\alpha }{\beta }}}\right)_{n}\left({{\frac {\beta }{\alpha }}}\right)_{n}=\prod _{{{\mathfrak {p}}|n\infty }}(\alpha ,\beta )_{{{\mathfrak {p}}}}\ .$

Notes

↑ Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.
↑ All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2
↑ Neukirch (1999) p. 336
↑ Neukirch (1999) p. 415

References

Gras, Georges (2003), Class field theory. From theory to practice, Springer Monographs in Mathematics, Berlin: Springer-Verlag, pp. 204–207, ISBN 3-540-44133-6, Zbl 1019.11032
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN 0-387-97329-X

Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer Science+Business Media, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021

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