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:
“A foreign minister, I will maintain it, can never be a good man of business if he is not an agreeable man of pleasure too. Half his business is done by the help of his pleasures: his views are carried on, and perhaps best, and most unsuspectedly, at balls, suppers, assemblies, and parties of pleasure; by intrigues with women, and connections insensibly formed with men, at those unguarded hours of amusement.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“We are the only class in history that has been left to fight its battles alone, unaided by the ruling powers. White labor and the freed black men had their champions, but where are ours?”
—Elizabeth Cady Stanton (1815–1902)
“Is not the tremendous strength in men of the impulse to creative work in every field precisely due to their feeling of playing a relatively small part in the creation of living beings, which constantly impels them to an overcompensation in achievement?”
—Karen Horney (1885–1952)
“We have our little theory on all human and divine things. Poetry, the workings of genius itself, which, in all times, with one or another meaning, has been called Inspiration, and held to be mysterious and inscrutable, is no longer without its scientific exposition. The building of the lofty rhyme is like any other masonry or bricklaying: we have theories of its rise, height, decline and fall—which latter, it would seem, is now near, among all people.”
—Thomas Carlyle (1795–1881)