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

In this context, the Hilbert class field of K is not just unramified at the finite places but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E.

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