Element to Power of Positive Characteristic of Field is Zero

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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 $\struct {F, +}$:

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


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:

$n \cdot a = 0$

$\blacksquare$


Sources