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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.

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 the Hilbert class field of $Q(\sqrt{-d})$, therefore $Q(\sqrt{-d})$ isn't an UFD.

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.

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Hilbert class field of Quadratic fields

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 the Hilbert class field of $Q(\sqrt{-d})$, therefore $Q(\sqrt{-d})$ isn't an UFD.