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Hilbert class field of Quadratic fields

MathOverflow Asked by Sina on February 14, 2021

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.

One 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

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