Definition:Power of Element/Field

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

Let $r \in R^*$ where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$.

Let $n \in \Z$ be an integer.

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

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

The definition of $n$th power of $r$ in $R$ as the the $n$th power of $r$ with respect to the group $\struct {R^*, \circ}$ can be extended to $0_R$ for positive values of $n$.

For all $n \in \Z_{\ge 0}$ the $n$th power of $0_R$ in $R$ is defined:
 * $\paren{0_R}^n = \begin {cases}

1_R & : n = 0 \\ 0_R & : n > 0 \end {cases}$

It should be noted that for all $n < 0$ the $n$th power of $0_R$ is not defined.

Also see

 * User:Leigh.Samphier/P-adicNumbers/Index Laws for Division Ring


 * User:Leigh.Samphier/P-adicNumbers/Index Laws/Negative Index/Division Ring


 * User:Leigh.Samphier/P-adicNumbers/Index Laws/Sum of Indices/Division Ring


 * User:Leigh.Samphier/P-adicNumbers/Index Laws/Product of Indices/Division Ring