multiplicative congruence

Let ${\mathfrak{p}}$ be any real prime of a number field $K$ , and write $i:K\longrightarrow\mathbb{R}$ for the corresponding real embedding of $K$ . We say two elements $\alpha,\beta\in K$ are multiplicatively congruent mod ${\mathfrak{p}}$ if the real numbers $i(\alpha)$ and $i(\beta)$ are either both positive or both negative.

Now let ${\mathfrak{p}}$ be a finite prime of $K$ , and write $(\mathcal{O}_{K})_{\mathfrak{p}}$ for the localization of the ring of integers $\mathcal{O}_{K}$ of $K$ at ${\mathfrak{p}}$ . For any natural number $n$ , we say $\alpha$ and $\beta$ are multiplicatively congruent mod ${\mathfrak{p}}^{n}$ if they are members of the same coset of the subgroup $1+{\mathfrak{p}}^{n}(\mathcal{O}_{K})_{\mathfrak{p}}$ of the multiplicative group $K^{\times}$ of $K$ .

If ${\mathfrak{m}}$ is any modulus for $K$ , with factorization

${\mathfrak{m}}=\prod_{{\mathfrak{p}}}{\mathfrak{p}}^{n_{\mathfrak{p}}},$

then we say $\alpha$ and $\beta$ are multiplicatively congruent mod ${\mathfrak{m}}$ if they are multiplicatively congruent mod ${\mathfrak{p}}^{n_{\mathfrak{p}}}$ for every prime ${\mathfrak{p}}$ appearing in the factorization of ${\mathfrak{m}}$ .

Multiplicative congruence of $\alpha$ and $\beta$ mod ${\mathfrak{m}}$ is commonly denoted using the notation

$\alpha\equiv^{*}\beta\pmod{{\mathfrak{m}}}.$

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