Ideal Class Group - Connections To Class Field Theory

Connections To Class Field Theory

Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. The Hilbert class field L of a number field K is unique and has the following properties:

  • Every ideal of the ring of integers of K becomes principal in L, i.e., if I is an integral ideal of K then the image of I is a principal ideal in L.
  • L is a Galois extension of K with Galois group isomorphic to the ideal class group of K.

Neither property is particularly easy to prove.

Read more about this topic:  Ideal Class Group

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