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Hasse norm theorem
The Hasse norm theorem, Hasse's theorem on cyclic extensions, is a 1931 result of Helmut Hasse: in a cyclic extension of number fields, being a norm locally everywhere is equivalent to being a global norm.
Hilbert class field
The Hilbert class field of a number field K is its unique maximal unramified abelian extension, with Galois group isomorphic to K's ideal class group.
Principal ideal theorem
The principal ideal theorem, also called the Artin–Furtwängler theorem, states that every ideal of a number field becomes principal in its Hilbert class field, proved in 1930.
Unramified extension
An unramified extension is a field extension in which no prime of the base field acquires ramification, meaning the ramification index is 1 and residue fields extend separably.