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Hilbert Class Field


Given a number field K, there exists a unique maximal unramified Abelian extension L of K which contains all other unramified Abelian extensions of K. This finite field extension L is called the Hilbert class field of K. By a theorem of class field theory, the Galois group G=Gal(L/K) is isomorphic to the class group of K and for every subgroup G^' of G, there exists a unique unramified Abelian extension L^' of K contained in L such that G^'=Gal(L/L^').

The degree [L:K] of L over K is equal to the class number of K.


See also

Artin Map, Artin Symbol, Class Field, Class Field Theory, Class Number

This entry contributed by David Terr

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References

Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.Marcus, D. A. Number Fields, 3rd ed. New York: Springer-Verlag, 1996.

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Hilbert Class Field

Cite this as:

Terr, David. "Hilbert Class Field." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HilbertClassField.html

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