# ideal class

Let $K$ be a number field. Let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals in $\mathcal{O}_{K}$ (the ring of algebraic integers of $K$). Define a relation $\sim$ on the ideals of $\mathcal{O}_{K}$ in the following way: write $\mathfrak{a}\sim\mathfrak{b}$ if there exist nonzero elements $\alpha$ and $\beta$ of $\mathcal{O}_{K}$ such that $(\alpha)\mathfrak{a}=(\beta)\mathfrak{b}$.

The relation $\sim$ is an equivalence relation, and the equivalence classes under $\sim$ are known as .

The number of equivalence classes, denoted by $h$ or $h_{K}$, is called the 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 $\mathcal{O}_{K}$ in the following way.

Let $\mathfrak{a}$, $\mathfrak{b}$ be ideals of $\mathcal{O}_{K}$. Denote the ideal classes of which $\mathfrak{a}$ and $\mathfrak{b}$ are representatives by $[\mathfrak{a}]$ and $[\mathfrak{b}]$ respectively. Then define $\cdot$ by

 $[\mathfrak{a}]\cdot[\mathfrak{b}]=[\mathfrak{a}\mathfrak{b}]$

Let ${\cal C}=\{[\mathfrak{a}]\mid\mathfrak{a}\neq(0),\mathfrak{a}\text{ an ideal % of }\mathcal{O}_{K}\}$. With the above definition of multiplication, $\cal C$ 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