= Hilbert class field of Q of square root minus thirty
{c}
{title2=$H_{\mathbb Q(\sqrt{-30})}=\mathbb Q(\sqrt2,\sqrt{-3},\sqrt5)$}
The <ideal class group> of $\mathbb Q(\sqrt{-30})$ is $C_2\times C_2$. Its four reduced <binary quadratic forms> have coefficients $(1,0,30)$, $(2,0,15)$, $(3,0,10)$ and $(5,0,6)$. The displayed <multiquadratic field> has degree four over the base. The seven quadratic-subfield discriminants are $8,-3,5,-24,40,-15,-120$, whose product in <absolute value> is $120^4$. The <conductor-discriminant formula> and discriminant tower formula imply that the <relative discriminant> over $\mathbb Q(\sqrt{-30})$ is the <unit ideal>. This unramified abelian degree-four extension is therefore its <Hilbert class field>.
Back to article page
