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.
Note that in this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
Read more about Hilbert Class Field: Examples, History, Additional Properties, Explicit Constructions, Generalizations
Famous quotes containing the words class and/or field:
“The traveler to the United States will do well ... to prepare himself for the class-consciousness of the natives. This differs from the already familiar English version in being more extreme and based more firmly on the conviction that the class to which the speaker belongs is inherently superior to all others.”
—John Kenneth Galbraith (b. 1908)
“We need a type of theatre which not only releases the feelings, insights and impulses possible within the particular historical field of human relations in which the action takes place, but employs and encourages those thoughts and feelings which help transform the field itself.”
—Bertolt Brecht (1898–1956)