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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. 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. Let k be an algebraic number field with ring of integers O k {displaystyle {mathcal {O}}_{k}} that contains a primitive n-th root of unity ζ n . {displaystyle zeta _{n}.} Let p ⊂ O k {displaystyle {mathfrak {p}}subset {mathcal {O}}_{k}} be a prime ideal and assume that n and p {displaystyle {mathfrak {p}}} are coprime (i.e. n ∉ p {displaystyle n ot in {mathfrak {p}}} .) The norm of p {displaystyle {mathfrak {p}}} is defined as the cardinality of the residue class ring (note that since p {displaystyle {mathfrak {p}}} is prime the residue class ring is a finite field): An analogue of Fermat's theorem holds in O k . {displaystyle {mathcal {O}}_{k}.} If α ∈ O k − p , {displaystyle alpha in {mathcal {O}}_{k}-{mathfrak {p}},} then And finally, suppose N p ≡ 1 mod n . {displaystyle mathrm {N} {mathfrak {p}}equiv 1{mod {n}}.} These facts imply that is well-defined and congruent to a unique n {displaystyle n} -th root of unity ζ n s . {displaystyle zeta _{n}^{s}.} This root of unity is called the n-th power residue symbol for O k , {displaystyle {mathcal {O}}_{k},} and is denoted by The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol ( ζ {displaystyle zeta } is a fixed primitive n {displaystyle n} -th root of unity):

[ "Isotropic quadratic form", "Binary quadratic form", "Quadratic field" ]
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