Power residue symbol – Wikipedia

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[edit]

Let k be an algebraic number field with ring of integers

${displaystyle {mathcal {O}}_{k}}$

that contains a primitive n-th root of unity

${displaystyle zeta _{n}.}$

Let

${displaystyle {mathfrak {p}}subset {mathcal {O}}_{k}}$

be a prime ideal and assume that n and

${displaystyle {mathfrak {p}}}$

are coprime (i.e.

${displaystyle nnot in {mathfrak {p}}}$

.)

The norm of

${displaystyle {mathfrak {p}}}$

is defined as the cardinality of the residue class ring (note that since

${displaystyle {mathfrak {p}}}$

is prime the residue class ring is a finite field):

${displaystyle mathrm {N} {mathfrak {p}}:=|{mathcal {O}}_{k}/{mathfrak {p}}|.}$

An analogue of Fermat’s theorem holds in

${displaystyle {mathcal {O}}_{k}.}$

If

${displaystyle alpha in {mathcal {O}}_{k}-{mathfrak {p}},}$

then

${displaystyle alpha ^{mathrm {N} {mathfrak {p}}-1}equiv 1{bmod {mathfrak {p}}}.}$

And finally, suppose

${displaystyle mathrm {N} {mathfrak {p}}equiv 1{bmod {n}}.}$

These facts imply that

${displaystyle alpha ^{frac {mathrm {N} {mathfrak {p}}-1}{n}}equiv zeta _{n}^{s}{bmod {mathfrak {p}}}}$

is well-defined and congruent to a unique

${displaystyle n}$

-th root of unity

${displaystyle zeta _{n}^{s}.}$

Definition[edit]

This root of unity is called the n-th power residue symbol for

${displaystyle {mathcal {O}}_{k},}$

and is denoted by

${displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}=zeta _{n}^{s}equiv alpha ^{frac {mathrm {N} {mathfrak {p}}-1}{n}}{bmod {mathfrak {p}}}.}$

Properties[edit]

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

${displaystyle zeta }$

is a fixed primitive

${displaystyle n}$

-th root of unity):

${displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}={begin{cases}0&alpha in {mathfrak {p}}\1&alpha not in {mathfrak {p}}{text{ and }}exists eta in {mathcal {O}}_{k}:alpha equiv eta ^{n}{bmod {mathfrak {p}}}\zeta &alpha not in {mathfrak {p}}{text{ and there is no such }}eta end{cases}}}$

In all cases (zero and nonzero)

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

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

${displaystyle alpha equiv beta {bmod {mathfrak {p}}}quad Rightarrow quad left({frac {alpha }{mathfrak {p}}}right)_{n}=left({frac {beta }{mathfrak {p}}}right)_{n}}$

Relation to the Hilbert symbol[edit]

The n-th power residue symbol is related to the Hilbert symbol

${displaystyle (cdot ,cdot )_{mathfrak {p}}}$

for the prime

${displaystyle {mathfrak {p}}}$

by

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

in the case

${displaystyle {mathfrak {p}}}$

coprime to n, where

${displaystyle pi }$

is any uniformising element for the local field

${displaystyle K_{mathfrak {p}}}$

.^[3]

Generalizations[edit]

The

${displaystyle n}$

-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

${displaystyle {mathfrak {a}}subset {mathcal {O}}_{k}}$

is the product of prime ideals, and in one way only:

${displaystyle {mathfrak {a}}={mathfrak {p}}_{1}cdots {mathfrak {p}}_{g}.}$

The

${displaystyle n}$

-th power symbol is extended multiplicatively:

${displaystyle left({frac {alpha }{mathfrak {a}}}right)_{n}=left({frac {alpha }{{mathfrak {p}}_{1}}}right)_{n}cdots left({frac {alpha }{{mathfrak {p}}_{g}}}right)_{n}.}$

For

${displaystyle 0neq beta in {mathcal {O}}_{k}}$

then we define

${displaystyle left({frac {alpha }{beta }}right)_{n}:=left({frac {alpha }{(beta )}}right)_{n},}$

where

${displaystyle (beta )}$

is the principal ideal generated by

${displaystyle beta .}$

Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.

• If
  ${displaystyle alpha equiv beta {bmod {mathfrak {a}}}}$

  then

  ${displaystyle left({tfrac {alpha }{mathfrak {a}}}right)_{n}=left({tfrac {beta }{mathfrak {a}}}right)_{n}.}$

  

• 
  ${displaystyle left({tfrac {alpha }{mathfrak {a}}}right)_{n}left({tfrac {beta }{mathfrak {a}}}right)_{n}=left({tfrac {alpha beta }{mathfrak {a}}}right)_{n}.}$

  

• 
  ${displaystyle left({tfrac {alpha }{mathfrak {a}}}right)_{n}left({tfrac {alpha }{mathfrak {b}}}right)_{n}=left({tfrac {alpha }{mathfrak {ab}}}right)_{n}.}$

  

Since the symbol is always an

${displaystyle n}$

-th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an

${displaystyle n}$

-th power; the converse is not true.

Power reciprocity law[edit]

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

${displaystyle left({frac {alpha }{beta }}right)_{n}left({frac {beta }{alpha }}right)_{n}^{-1}=prod _{{mathfrak {p}}|ninfty }(alpha ,beta )_{mathfrak {p}},}$

whenever

${displaystyle alpha }$

and

${displaystyle beta }$

are coprime.

See also[edit]

References[edit]

• 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, Springer Monographs in Mathematics, 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, vol. 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021