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:

Also see

 * Index Laws for Monoid
 * Powers of Group Elements