multiplicative congruence

Let 𝔭 be any real prime of a number fieldMathworldPlanetmath K, and write i:K for the corresponding real embedding of K. We say two elements α,βK are multiplicatively congruent mod 𝔭 if the real numbers i(α) and i(β) are either both positive or both negative.

Now let 𝔭 be a finite prime of K, and write (𝒪K)𝔭 for the localization of the ring of integersMathworldPlanetmath 𝒪K of K at 𝔭. For any natural number n, we say α and β are multiplicatively congruent mod 𝔭n if they are members of the same coset of the subgroupMathworldPlanetmathPlanetmath 1+𝔭n(𝒪K)𝔭 of the multiplicative groupMathworldPlanetmath K× of K.

If 𝔪 is any modulusMathworldPlanetmathPlanetmathPlanetmath for K, with factorization


then we say α and β are multiplicatively congruent mod 𝔪 if they are multiplicatively congruent mod 𝔭n𝔭 for every prime 𝔭 appearing in the factorization of 𝔪.

Multiplicative congruence of α and β mod 𝔪 is commonly denoted using the notation

Title multiplicative congruence
Canonical name MultiplicativeCongruence
Date of creation 2013-03-22 12:50:16
Last modified on 2013-03-22 12:50:16
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 11R37
Synonym multiplicatively congruent
Related topic Congruence2