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