Index Laws for Field

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 the following hold:

$(a):\quad \forall a, b \in \F^* : \forall n \in \Z : a^n \circ b^n = \paren{ab}^n$

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

$\forall a \in \F^* : \forall n \in \Z : \paren{a^n}^{-1} = a^{-n} = \paren{a^{-1}}^n$

$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : a^m \circ a^n = a^\paren{m + n}$

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

$(a):\quad \forall a \in \F^* : \forall n, m \in \Z : \paren{a^m}^n = a^\paren{mn}$

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

The name index laws originates from the name index to describe the exponent $y$ in the power $x^y$.