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Takagi existence theorem
In class field theory, the Takagi existence theorem states in part that if K is a number field with class group G, there exists a unique abelian extension L/K with Galois group G, such that every ideal in K becomes principal in L, and that L is characterized by the property that it is the maximal unramified abelian extension of K. The theorem tells us that the Hilbert class field conjectured by Hilbert always exists, but it required Artin and Furtwängler to prove that principalization occurs.
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