Element to Power of Positive Characteristic of Field is Zero

Theorem
Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.

Let the characteristic of $F$ be $n$ such that $n > 0$.

Then:
 * $n \cdot a = 0$

where $n \cdot a$ denotes the power of $a$ in the context of the additive group $\struct {F, +}$:
 * $n \cdot a = \begin {cases}

0 & : n = 0 \\ \paren {n - 1} a + a & : n > 0 \end {cases}$

Proof
By definition, the characteristic of $\struct {F, +, \times}$ is the order of the additive group $\struct {F, +}$.

By Element to Power of Group Order is Identity it follows directly that:
 * $n \cdot a = 0$