Definition:Class Group

Definition

Let $F$ be a field of algebraic numbers.

Let $H$ be the set of ideals of the elements of $F$.

Let $\RR$ be the equivalence relation on $H$ defined as:

$\forall I, J \in H: I \mathrel \RR J \iff \exists S, T, \in H: I S = J T$

where $S$ and $T$ are principal ideals of $F$.

Let $G$ be the group of equivalence classes of $\RR$ such that the group operation is that the product of the equivalence classes containing $I$ and $J$ is the equivalence class containing $I J$.

This group is known as the class group of $F$.

Class Number

Let $G$ be the class group of $F$.

The class number of $F$ is the number of elements of $G$.

Also see

• Results about class groups can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): class group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): class group