multiplicative congruenceMultiplicativeCongruence2002-07-11 11:27:072002-07-11 11:27:07Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\ilim}{\,\underset{\longleftarrow}{\lim}\,}Let $\p$ be any real prime of a number field $K$, and write $i: K \lra \R$ for the corresponding real embedding of $K$. We say two elements $\alpha, \beta \in K$ are {\em multiplicatively congruent} mod $\p$ if the real numbers $i(\alpha)$ and $i(\beta)$ are either both positive or both negative.
Now let $\p$ be a finite prime of $K$, and write $(\O_K)_\p$ for the localization of the ring of integers $\O_K$ of $K$ at $\p$. For any natural number $n$, we say $\alpha$ and $\beta$ are {\em multiplicatively congruent} mod $\p^n$ if they are members of the same coset of the subgroup $1+\p^n(\O_K)_\p$ of the multiplicative group $K^\times$ of $K$.
If $\m$ is any modulus for $K$, with factorization
$$
\m = \prod_{\p} \p^{n_\p},
$$
then we say $\alpha$ and $\beta$ are {\em multiplicatively congruent} mod $\m$ if they are multiplicatively congruent mod $\p^{n_\p}$ for every prime $\p$ appearing in the factorization of $\m$.
Multiplicative congruence of $\alpha$ and $\beta$ mod $\m$ is commonly denoted using the notation
$$
\alpha \equiv^* \beta \pmod{\m}.
$$