Hilbert class field

In algebraic number theory, the Hilbert class field ${\displaystyle E}$ of a number field ${\displaystyle K}$ is the maximal abelian unramified extension of ${\displaystyle K}$. Its degree over ${\displaystyle K}$ equals the class number of ${\displaystyle K}$ and the Galois group of ${\displaystyle E}$ over ${\displaystyle K}$ is canonically isomorphic to the ideal class group of ${\displaystyle K}$ using Frobenius elements for prime ideals in ${\displaystyle K}$.

In this context, the Hilbert class field of ${\displaystyle K}$ is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ${\displaystyle K}$. That is, every real embedding of ${\displaystyle K}$ extends to a real embedding of ${\displaystyle E}$ (rather than to a complex embedding of ${\displaystyle E}$).