Hilbert 2-Class Field definition

Question

what is a Hilbert 2- class field? As a Hilbert Class field of a number field K is the maximal unramified abelian extension, of K,

Answer 1

Let be $ K^{(1)}$ the Hilbert Class field of a number field $K$ and $ K_p^{(1)}$ the Hilbert p-class field of $K$, then $ K_p^{(1)}$ is exactly the sub extension of $ K^{(1)} / K$ which corresponds (by Galois correspondence) to Galois sub group of $Gal( K^{(1)} / K)$ which is the direct product of all the $ q$-Sylow $(q \neq p)$ . so is the sub extension of $ K^{(1)} / K$ whose Galois group $Gal( K_p^{(1)}/K)$ is isomorphic to the p-Sylow of $Gal( K^{(1)} / K)$.