Definition:Power of Element/Field

From ProofWiki
Jump to navigation Jump to search


Let $\struct {F, +, \circ}$ be a field with zero $0_F$ and unity $1_F$.

Let $a \in F^*$ where $F^*$ denotes the set of elements of $F$ without the zero $0_F$.

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

The $n$th power of $a$ in $F$ is defined as the $n$th power of $a$ with respect to the Abelian group $\struct {F^*, \circ}$:

$\forall n \in \Z: a^n = \begin {cases}

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

The definition of $n$th power of $a$ in $F$ as the the $n$th power of $a$ with respect to the monoid $\struct {F, \circ}$ can be extended to $0_F$ for positive values of $n$.

For all $n \in \Z_{\ge 0}$ the $n$th power of $0_F$ in $F$ is defined:

$\paren{0_F}^n = \begin {cases}

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

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

Also see