Examples
If the ring of integers of K is a unique factorization domain, in particular, if then K is its own Hilbert class field.
By contrast, let . By analyzing ramification degrees over, one can show that is an everywhere unramified extension of K, and it is certainly abelian. Hence the Hilbert class field of K is a nontrivial extension and the ring of integers of K cannot be a unique factorization domain. (In fact, using the Minkowski bound, one can show that K has class number exactly 2.) Hence, the Hilbert class field is .
To see why ramification at the archimedean primes must be taken into account, consider the real quadratic field K obtained by adjoining the square root of 3 to Q. This field has class number 1, but the extension K(i)/K is unramified at all prime ideals in K, so K admits finite abelian extensions of degree greater than 1 in which all primes of K are unramified. This doesn't contradict the Hilbert class field of K being K itself: every proper finite abelian extension of K must ramify at some place, and in the extension K(i)/K there is ramification at the archimedean places: the real embeddings of K extend to complex (rather than real) embeddings of K(i).
Read more about this topic: Hilbert Class Field
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)