quaternion algebra

A quaternion algebra over a field $K$ is a central simple algebra over $K$ which is four dimensional as a vector space over $K$ .

Examples:

•
For any field $K$ , the ring $M_{2\\times 2}(K)$ of $2\\times 2$ matrices with entries in $K$ is a quaternion algebra over $K$ . If $K$ is algebraically closed, then all quaternion algebras over $K$ are isomorphic to $M_{2\\times 2}(K)$ .
•
For $K=\\mathbb{R}$ , the well known algebra $\\mathbb{H}$ of Hamiltonian quaternions is a quaternion algebra over $\\mathbb{R}$ . The two algebras $\\mathbb{H}$ and $M_{2\\times 2}(\\mathbb{R})$ are the only quaternion algebras over $\\mathbb{R}$ , up to isomorphism.
•
When $K$ is a number field, there are infinitely many non–isomorphic quaternion algebras over $K$ . In fact, there is one such quaternion algebra for every even sized finite collection of finite primes or real primes of $K$ . The proof of this deep fact leads to many of the major results of class field theory.

One can show that every quaternion algebra over $K$ other than $M_{2\\times 2}(K)$ is always a division ring.

单词	QuaternionAlgebra
释义	quaternion algebra A quaternion algebra over a field $K$ is a central simple algebra over $K$ which is four dimensional as a vector space over $K$ . Examples: • For any field $K$ , the ring $M_{2\\times 2}(K)$ of $2\\times 2$ matrices with entries in $K$ is a quaternion algebra over $K$ . If $K$ is algebraically closed, then all quaternion algebras over $K$ are isomorphic to $M_{2\\times 2}(K)$ . • For $K=\\mathbb{R}$ , the well known algebra $\\mathbb{H}$ of Hamiltonian quaternions is a quaternion algebra over $\\mathbb{R}$ . The two algebras $\\mathbb{H}$ and $M_{2\\times 2}(\\mathbb{R})$ are the only quaternion algebras over $\\mathbb{R}$ , up to isomorphism. • When $K$ is a number field, there are infinitely many non–isomorphic quaternion algebras over $K$ . In fact, there is one such quaternion algebra for every even sized finite collection of finite primes or real primes of $K$ . The proof of this deep fact leads to many of the major results of class field theory. One can show that every quaternion algebra over $K$ other than $M_{2\\times 2}(K)$ is always a division ring.
随便看	UltraconnectedSpace ultraconnected space Ultrafactorial Ultrafactorial Ultrafilter ultrafilter Ultrametric ultrametric UltrametricSpace ultrametric space UltrametricTriangleInequality ultrametric triangle inequality Ultranet ultranet Ultrauniversal ultra-universal UnambiguityOfFactorialBaseRepresentation unambiguity of factorial base representation UncertaintyPrinciple uncertainty principle UncertaintyTheorem uncertainty theorem UnconditionalConvergence unconditional convergence Uncountable