PlanetMath: quaternion algebra

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quaternion algebra

(Definition)

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

Examples:

For any field , the ring $M_{2\times 2}(K)$ of $2\times 2$ matrices with entries in is a quaternion algebra over . If is algebraically closed, then all quaternion algebras over 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 is a number field, there are infinitely many non-isomorphic quaternion algebras over . In fact, there is one such quaternion algebra for every even sized finite collection of finite primes or real primes of . 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 other than $M_{2\times 2}(K)$ is always a division ring.

"quaternion algebra" is owned by djao.

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AMS MSC:	16K20 (Associative rings and algebras :: Division rings and semisimple Artin rings :: Finite-dimensional)
	11R52 (Number theory :: Algebraic number theory: global fields :: Quaternion and other division algebras: arithmetic, zeta functions)

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