# ideal class

Let $K$ be a number field. Let $\mathrm{\pi \x9d\x94\x9e}$ and $\mathrm{\pi \x9d\x94\x9f}$ be ideals in ${\mathrm{\pi \x9d\x92\u037a}}_{K}$ (the ring of algebraic integers of $K$). Define a relation^{} $\beta \x88\u038c$ on the ideals of ${\mathrm{\pi \x9d\x92\u037a}}_{K}$ in the following way: write $\mathrm{\pi \x9d\x94\x9e}\beta \x88\u038c\mathrm{\pi \x9d\x94\x9f}$ if there exist nonzero elements $\mathrm{\Xi \pm}$ and $\mathrm{\Xi \xb2}$ of ${\mathrm{\pi \x9d\x92\u037a}}_{K}$ such that $(\mathrm{\Xi \pm})\beta \x81\u2019\mathrm{\pi \x9d\x94\x9e}=(\mathrm{\Xi \xb2})\beta \x81\u2019\mathrm{\pi \x9d\x94\x9f}$.

The relation $\beta \x88\u038c$ is an equivalence relation^{}, and the equivalence classes^{} under $\beta \x88\u038c$ are known as ideal classes^{}.

The number of equivalence classes, denoted by $h$ or ${h}_{K}$, is called the class number^{} of $K$.

Note that the set of ideals of any ring $R$ forms an abelian semigroup with the product of ideals as the semigroup operation^{}. By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of ${\mathrm{\pi \x9d\x92\u037a}}_{K}$ in the following way.

Let $\mathrm{\pi \x9d\x94\x9e}$, $\mathrm{\pi \x9d\x94\x9f}$ be ideals of ${\mathrm{\pi \x9d\x92\u037a}}_{K}$. Denote the ideal classes of which $\mathrm{\pi \x9d\x94\x9e}$ and $\mathrm{\pi \x9d\x94\x9f}$ are representatives by $[\mathrm{\pi \x9d\x94\x9e}]$ and $[\mathrm{\pi \x9d\x94\x9f}]$ respectively. Then define $\beta \x8b\x85$ by

$$[\mathrm{\pi \x9d\x94\x9e}]\beta \x8b\x85[\mathrm{\pi \x9d\x94\x9f}]=[\mathrm{\pi \x9d\x94\x9e}\beta \x81\u2019\mathrm{\pi \x9d\x94\x9f}]$$ |

Let $\mathrm{\pi \x9d\x92\x9e}=\{[\mathrm{\pi \x9d\x94\x9e}]\beta \x88\pounds \mathrm{\pi \x9d\x94\x9e}\beta \x89(0),\mathrm{\pi \x9d\x94\x9e}\beta \x81\u2019\text{\Beta an ideal of\Beta}\beta \x81\u2019{\mathrm{\pi \x9d\x92\u037a}}_{K}\}$.
With the above definition of multiplication^{}, $\mathrm{\pi \x9d\x92\x9e}$ is an abelian group^{}, called the ideal class group (or frequently just the *class group*) of $K$.

Note that the ideal class group of $K$ is simply the quotient group^{} of the ideal group of $K$ by the subgroup^{} of principal fractional ideals.

Title | ideal class |

Canonical name | IdealClass |

Date of creation | 2013-03-22 12:36:42 |

Last modified on | 2013-03-22 12:36:42 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 22 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11R29 |

Synonym | ideal classes |

Related topic | ExistenceOfHilbertClassField |

Related topic | FractionalIdeal |

Related topic | NumberField |

Related topic | UnramifiedExtensionsAndClassNumberDivisibility |

Related topic | ClassNumberDivisibilityInExtensions |

Related topic | PushDownTheoremOnClassNumbers |

Related topic | MinkowskisConstant |

Related topic | ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivis |

Defines | class number |

Defines | ideal class group |

Defines | class group |