Division Ring

Definition

Division Ring(skew field) $R$ is $1\ne 0$ and $R^{\times }=R-0$.

Example

Real

$\mathbb{H}=\left\{a+bi+cj+dk|a,b,c,d\in \mathbb{R}\right\}$ and $i^2=j^2=k^2=-1$. Using the fact that norm is multiplicative, ${\alpha }^{-1}=\bar{\alpha }/Norm(\alpha )$ for $\alpha \in \mathbb{H}$. Therefore ${\mathbb{H}}^{\times }=Q_8$ where $Q_8$ as a quaternion group, not a ring.