Hilbert class field of Quadratic fields

Question

Is there any method to find the Hilbert Class field of quadratic fields? Is there any bound for their dimensions? For example, if $4|d-1$ then $Q(sqrt{d},i)$ is contained in the Hilbert class field of $Q(sqrt{-d})$, therefore $Q(sqrt{-d})$ isn't an UFD.

Davood KHAJEHPOUR · Answer

For the case of imaginary quadratic fields $K=mathbb{Q}(sqrt{d_K})$, the Hilbert class field can be given as $Kleft(j(frac{d_k+sqrt{d_k}}{2})right)$, where $j$ is the $j$-invariant. This is just a theoretical answer; and note that computing $j(frac{d_k+sqrt{d_k}}{2})$ in practice, is not easy.

Answered by Davood KHAJEHPOUR on February 14, 2021

Hilbert class field of Quadratic fields

One Answer

Add your own answers!

Ask a Question