nLab
quaternion

Idea

The quaternions form the largest associative normed division algebra, usually denoted $ℍ$ after William Rowan Hamilton? (since $\mathbb{Q}$ is taken for the rational numbers).

Normed division algebra structure

Concretely, the structure of $ℍ$ as an $ℝ$-algebra is given by a basis $\{1,i,j,k\}$ of the underlying vector space of $ℍ$, equipped with a multiplication table where $1$ is the identity element and otherwise uniquely specified by the equations

$$i^2=j^2=k^2=ijk=-1,$$i^2 = j^2 = k^2 = i j k = -1,

and extended by $ℝ$-linearity to all of $ℍ$. The norm on $ℍ$ is given by

$${\parallel \alpha \parallel }^2=\alpha \overline{\alpha }$${\|\alpha\|}^2 = \alpha \widebar{\alpha}

where given an $ℝ$-linear combination $\alpha =a1+bi+cj+dk$, we define $\overline{\alpha }≔a1-bi-cj-dk$. A simple calculation yields

$${\parallel \alpha \parallel }^2=a^2+b^2+c^2+d^2$${\|\alpha\|}^2 = a^2 + b^2 + c^2 + d^2

whence for $\alpha \ne 0$, the multiplicative inverse is

$${\alpha }^{-1}=\frac{1}{{\parallel \alpha \parallel }^2}\overline{\alpha }.$$\alpha^{-1} = \frac1{{\|\alpha\|}^2} \widebar{\alpha}.

In this way $ℍ$ is a normed division algebra.

References

A survey is in

• T. Y. Lam, Hamilton’s Quaternions (ps)