Application
Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that generate the elements of order two in the Brauer group of F. (For some fields, including algebraic number fields, every element of index 2 in its Brauer group is represented by a quaternion algebra. A theorem of Merkurjev says the elements of index 2 in the Brauer group of any field are represented by a tensor product of quaternion algebras.) In particular, over p-adic fields the construction of quaternion algebras can be viewed as the quadratic Hilbert symbol of local class field theory.
Read more about this topic: Quaternion Algebra
Famous quotes containing the word application:
““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“We will not be imposed upon by this vast application of forces. We believe that most things will have to be accomplished still by the application called Industry. We are rather pleased, after all, to consider the small private, but both constant and accumulated, force which stands behind every spade in the field. This it is that makes the valleys shine, and the deserts really bloom.”
—Henry David Thoreau (1817–1862)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)