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 higherreciprocity laws.

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. ${\displaystyle n\not \in {\mathfrak {p}}}$.)

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