# Powers of Field Elements Commute

## Theorem

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

Let $F^* = F \setminus {0_F}$ denote the set of elements of $F$ without the zero $0_F$.

Then:

$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^n \circ a^m$
$(b):\quad \forall a \in F : \forall n, m \in \Z_{\ge 0} : a^m \circ a^n = a^n \circ a^m$

## Proof

### Statement $(a)$

By Definition of Field:

$\struct{F^*, \circ}$ is an Abelian group

By Definition of Power of Field Element:

For all $a \in F^*$ and $n \in \Z$, $a^n$ is defined as the $n$th power of $a$ with respect to the Abelian group $\struct {F^*, \circ}$
$\forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^n \circ a^m$

$\Box$

### Statement $(b)$

Let $m,n \in \Z_{\ge 0}$ be arbitrary elements of $\Z_{\ge 0}$.

For $a \in F^*$, $(b)$ follows from $(a)$.

It remains to show that $(b)$ holds for $0_F$.

#### Case 1: $m = 0$

Let $m = 0$.

We have:

 $\ds \paren{0_F}^m \circ \paren{0_F}^n$ $=$ $\ds \paren{0_F}^0 \circ \paren{0_F}^n$ $\ds$ $=$ $\ds 1_F \circ \paren{0_F}^n$ Definition of Power of Field Element $\ds$ $=$ $\ds \paren{0_F}^n$ Field Axiom $\text M3$: Identity for Product $\ds$ $=$ $\ds \paren{0_F}^n \circ 1_F$ Field Axiom $\text M3$: Identity for Product $\ds$ $=$ $\ds \paren{0_F}^n \circ \paren{0_F}^0$ Definition of Power of Field Element $\ds$ $=$ $\ds \paren{0_F}^n \circ \paren{0_F}^m$

$\Box$

#### Case 2: $m \ne 0$

Let $m \ne 0$.

We have:

 $\ds \paren{0_F}^m \circ \paren{0_F}^n$ $=$ $\ds 0_F \circ \paren{0_F}^n$ Definition of Power of Field Element $\ds$ $=$ $\ds 0_F$ Field Product with Zero $\ds$ $=$ $\ds \paren{0_F}^n \circ 0_F$ Field Product with Zero $\ds$ $=$ $\ds \paren{0_F}^n \circ \paren{0_F}^m$ Definition of Power of Field Element

$\Box$

In both cases:

$\paren{0_F}^m \circ \paren{0_F}^n = \paren{0_F}^n \circ \paren{0_F}^m$

Since $m,n$ were arbitrary elements of $\Z_{\ge 0}$:

$\forall n, m \in \Z_{\ge 0} : \paren{0_F}^m \circ \paren{0_F}^n = \paren{0_F}^n \circ \paren{0_F}^m$

$\blacksquare$