# Integral Domain with Characteristic Zero

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## Theorem

In an integral domain with characteristic zero, every non-zero element has infinite order under ring addition.

## Proof

Let $\struct {D, +, \circ}$ be an integral domain, whose zero is $0_D$ and whose unity is $1_D$, such that $\Char D = 0$.

Let $x \in D, x \ne 0_D$.

Then:

\(\displaystyle \forall n \in \Z_{>0}: \ \ \) | \(\displaystyle n \cdot x\) | \(=\) | \(\displaystyle n \cdot \paren {x \circ 1_D}\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {n \circ 1_D} \cdot x\) | Powers of Ring Elements |

Then:

\(\displaystyle x\) | \(\ne\) | \(\displaystyle 0_D\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle n \cdot 1_D\) | \(\ne\) | \(\displaystyle 0_D\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle n \cdot x\) | \(\ne\) | \(\displaystyle 0_D\) | Definition of Integral Domain |

That is, $x$ has infinite order in $\struct {D, +}$.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 61.2$ Characteristic of an integral domain or field