Tuesday, May 29, 2018

Methods to compute the class number

The class number is an important term from algebraic number theory. It specifies the order of the class group of an algebraic number field $K$ and can be interpreted as the number of different ways to factorize an element of $K$ into prime elements. If the class number is equal to $1$, the $K$ is called a unique factorization domain, e.g. the integers $\mathbb{Z}$. It is long known that for imaginary quadratic fields the class number is $1$ only for the values: $$K = \mathbf{Q}(\sqrt{d}), d \in \{-1,-2,-3,-7,-11,-19,-43,-67,-163\}$$ There are a surprisingly large number of ways to compute the class number of a quadratic field, which i want to show you next.