Class field theory

Hilbert class field

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 (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). (Wikipedia).

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Related pages

Prime ideal | Ray class field | Class field theory | Ideal (ring theory) | Modulus (algebraic number theory) | Ring of integers | Automorphism | Complex multiplication | Philipp Furtwängler | Field extension | Field (mathematics) | Real number | Algebraic number theory | Principal ideal theorem | J-invariant | Galois group | Prime number | Discriminant of an algebraic number field | Elliptic curve | Quadratic field | Unique factorization domain | Principal ideal | Ideal class group