# multiplicative congruence

Let $\U0001d52d$ be any real prime of a number field^{} $K$, and write $i:K\u27f6\mathbb{R}$ for the corresponding real embedding of $K$. We say two elements $\alpha ,\beta \in K$ are multiplicatively congruent mod $\U0001d52d$ if the real numbers $i(\alpha )$ and $i(\beta )$ are either both positive or both negative.

Now let $\U0001d52d$ be a finite prime of $K$, and write ${({\mathcal{O}}_{K})}_{\U0001d52d}$ for the localization of the ring of integers^{} ${\mathcal{O}}_{K}$ of $K$ at $\U0001d52d$. For any natural number $n$, we say $\alpha $ and $\beta $ are multiplicatively congruent mod ${\U0001d52d}^{n}$ if they are members of the same coset of the subgroup^{} $1+{\U0001d52d}^{n}{({\mathcal{O}}_{K})}_{\U0001d52d}$ of the multiplicative group^{} ${K}^{\times}$ of $K$.

If $\U0001d52a$ is any modulus^{} for $K$, with factorization

$$\U0001d52a=\prod _{\U0001d52d}{\U0001d52d}^{{n}_{\U0001d52d}},$$ |

then we say $\alpha $ and $\beta $ are multiplicatively congruent mod $\U0001d52a$ if they are multiplicatively congruent mod ${\U0001d52d}^{{n}_{\U0001d52d}}$ for every prime $\U0001d52d$ appearing in the factorization of $\U0001d52a$.

Multiplicative congruence of $\alpha $ and $\beta $ mod $\U0001d52a$ is commonly denoted using the notation

$$\alpha {\equiv}^{*}\beta \phantom{\rule{veryverythickmathspace}{0ex}}(mod\U0001d52a).$$ |

Title | multiplicative congruence |
---|---|

Canonical name | MultiplicativeCongruence |

Date of creation | 2013-03-22 12:50:16 |

Last modified on | 2013-03-22 12:50:16 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11R37 |

Synonym | multiplicatively congruent |

Related topic | Congruence2 |