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:
“The connections between and among women are the most feared, the most problematic, and the most potentially transforming force on the planet.”
—Adrienne Rich (b. 1929)
“The great disadvantage, and advantage, of the small urban bourgeois is his limited outlook. He sees the world as a middle- class world, and everything outside these limits is either laughable or slightly wicked.”
—George Orwell (1903–1950)
“The snow had begun in the gloaming,
And busily all the night
Had been heaping field and highway
With a silence deep and white.”
—James Russell Lowell (1819–1891)
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)