PlanetMath: multiplicative congruence

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multiplicative congruence

(Definition)

Let ${\mathfrak{p}}$ be any real prime of a number field , and write $i: K \longrightarrow \mathbb{R}$ for the corresponding real embedding of . 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 , and write $(\O _K)_{\mathfrak{p}}$ for the localization of the ring of integers $\O _K$ of at ${\mathfrak{p}}$ . For any natural number , we say $\alpha$ and $\beta$ are multiplicatively congruent mod ${\mathfrak{p}}^n$ if they are members of the same coset of the subgroup $1+{\mathfrak{p}}^n(\O _K)_{\mathfrak{p}}$ of the multiplicative group $K^\times$ of .

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

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

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

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

"multiplicative congruence" is owned by djao.

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