# Integral Domain with Characteristic Zero

## 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 $\struct {D, +}$.

$\blacksquare$