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}}}.