# 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 MultiplicativeCongruence 2013-03-22 12:50:16 2013-03-22 12:50:16 djao (24) djao (24) 4 djao (24) Definition msc 11R37 multiplicatively congruent Congruence2