Additional Properties
The Hilbert class field E also satisfies the following:
- E is a finite Galois extension of K and =hK, where hK is the class number of K.
- The ideal class group of K is isomorphic to the Galois group of E over K.
- Every ideal of OK is a principal ideal of the ring extension OE (principal ideal theorem).
- Every prime ideal P of OK decomposes into the product of hK/f prime ideals in OE, where f is the order of in the ideal class group of OK.
In fact, E is the unique field satisfying the first, second, and fourth properties.
Read more about this topic: Hilbert Class Field
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