Associative rings and algebras Division rings and central simple algebras Example

The quaternions

Example

The quaternions $\mathbb{H}$ are the real vector space with basis $1, i, j, k$ . Each quaternion is $a + b i + c j + d k$ with $a, b, c, d \in \mathbb{R}$ . The multiplication is $\mathbb{R}$ -bilinear, $1$ is the identity, and the basis elements obey $i^2 = j^2 = k^2 = i j k = -1.$ These rules give $i j = k = -j i$ , $j k = i = -k j$ and $k i = j = -i k$ . With this multiplication, $\mathbb{H}$ is a division ring that is not commutative.

Remarks

The conjugate of $q = a + b i + c j + d k$ is $\bar{q} = a - b i - c j - d k$ , and $q \bar{q} = a^2 + b^2 + c^2 + d^2$ . So a nonzero $q$ has the inverse $q^{-1} = \bar{q} / (a^2 + b^2 + c^2 + d^2)$ . The quaternions are not a field, because $i j \neq j i$ .

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Group concept
Ring definition
Unit of a ring definition
Division ring definition
The quaternions example

Tag	`0GE4DVQKR8`
URI	`m2://0GE4DVQKR8`
Aliases	`hamilton-quaternions`
Kind	example
Writing	W2 usable
MSC	16K20
Instance of	Division ring
Uses	Division ring
Same as	Wikidata Q173853 Wikipedia Quaternion

The quaternions

Example

Remarks

Reading path

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