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multiplicative congruence (Definition)

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 $ (\O _K)_{\mathfrak{p}}$ for the localization of the ring of integers $ \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(\O _K)_{\mathfrak{p}}$ of the multiplicative group $ K^\times$ of $ K$.

If $ {\mathfrak{m}}$ is any modulus for $ K$, 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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See Also: congruence

Other names:  multiplicatively congruent
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Cross-references: prime, modulus, multiplicative group, subgroup, coset, natural number, ring of integers, localization, finite prime, negative, positive, real numbers, real embedding, number field, real prime
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This is version 1 of multiplicative congruence, born on 2002-07-11.
Object id is 3163, canonical name is MultiplicativeCongruence.
Accessed 3155 times total.

Classification:
AMS MSC11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)

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