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
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 89 \alpha$