Definition:Power of Element/Field

Definition
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.

Let $R^* = R \setminus \set{0_R}$.

Let $r \in R$.

Let $n \in \Z_{\ge 0}$ be the set of positive integers.

The $n$th power of $r$ in $R$ is defined as the $n$th power of $r$ with respect to the ring with unity $\struct {R, +, \circ}$:
 * $\forall n \in \Z_{\ge 0}: r^n = \begin {cases}

1_R & : n = 0 \\ r^{n - 1} \circ r & : n > 0 \end {cases}$

The definition of $n$th power of $r$ in $R$ can be extended to allow negative values of $n$ on $R^*$.

Let $r \in R^*$.

Let $n \in \Z_{< 0}$ be the set of strictly negative integers.

The $n$th power of $r$ in $R$ is defined as the $m$th power of $s$ with respect to the group $\struct {R^*, \circ}$:
 * $\forall n \in \Z_{< 0}: r^n = \paren{r^{-1}}^{-n}$

It should be noted that the $m$th power of $0_R$ is not defined.

Also see

 * User:Leigh.Samphier/P-adicNumbers/Powers of Division Ring Elements