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:
\(\ds \forall n \in \Z_{>0}: \, \) | \(\ds n \cdot x\) | \(=\) | \(\ds n \cdot \paren {x \circ 1_D}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {n \circ 1_D} \cdot x\) | Integral Multiple of Ring Element |
Then:
\(\ds x\) | \(\ne\) | \(\ds 0_D\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \cdot 1_D\) | \(\ne\) | \(\ds 0_D\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \cdot x\) | \(\ne\) | \(\ds 0_D\) | 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