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
Famous quotes containing the words connections, class, field and/or theory:
“... feminism is a political term and it must be recognized as such: it is political in women’s terms. What are these terms? Essentially it means making connections: between personal power and economic power, between domestic oppression and labor exploitation, between plants and chemicals, feelings and theories; it means making connections between our inside worlds and the outside world.”
—Anica Vesel Mander, U.S. author and feminist, and Anne Kent Rush (b. 1945)
“Mankind divides itself into two classes,—benefactors and malefactors. The second class is vast, the first a handful.”
—Ralph Waldo Emerson (1803–1882)
“What though the field be lost?
All is not lost; the unconquerable Will,
And study of revenge, immortal hate,
And courage never to submit or yield:
And what is else not to be overcome?”
—John Milton (1608–1674)
“There never comes a point where a theory can be said to be true. The most that one can claim for any theory is that it has shared the successes of all its rivals and that it has passed at least one test which they have failed.”
—A.J. (Alfred Jules)