Earlier Work
A special case of the existence theorem is when m = 1 and H = P1. In this case the generalized ideal class group is the ideal class group of K, and the existence theorem says there exists a unique abelian extension L/K with Galois group isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is called the Hilbert class field. It was conjectured by David Hilbert to exist, and existence in this special case was proved by Furtwängler in 1907, before Takagi's general existence theorem.
A further and special property of the Hilbert class field, not true of other abelian extensions of a number field, is that all ideals in a number field become principal in the Hilbert class field. It required Artin and Furtwängler to prove that principalization occurs.
Read more about this topic: Takagi Existence Theorem
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